Declaration of Authorship

I, Shivani Goyal , declare that the work presented in it are my own except for others work which is clearly referenced.

Signed: Shivani Goyal
Date: May 24,2020

PART 1 - TIME SERIES ANALYSIS

Loading and installing all the Required packages

library(readr)
library(tidyr)
library(ggplot2)
library(pastecs)
library(dplyr)
library(VIM)
library(corrplot)
library(imputeTS)
library(lubridate)
library(forecast)
library(tseries)

1 - Reading the Assignment Dataset. The provided data file is excel file. The structure of the data is corrected by following given steps below:-

  1. Reading the Assignment Dataset
Assignment_Data <- read_csv("E:/shivani_timeseries/Assignment Data.csv", col_names = FALSE)
Assignment_Data<- as.data.frame(Assignment_Data)
head(Assignment_Data)
##         X1  X2  X3  X4  X5  X6  X7  X8  X9 X10 X11 X12 X13 X14 X15 X16 X17 X18
## 1 1/1/1998 0.8 1.8 2.4 2.1   2 2.1 1.5 1.7 1.9 2.3 3.7 5.5 5.1 5.4 5.4 4.7 4.3
## 2 1/2/1998 2.8 3.2 3.3 2.7 3.3 3.2 2.9 2.8 3.1 3.4 4.2 4.5 4.5 4.3 5.5 5.1 3.8
## 3 1/3/1998 2.9 2.8 2.6 2.1 2.2 2.5 2.5 2.7 2.2 2.5 3.1   4 4.4 4.6 5.6 5.4 5.2
## 4 1/4/1998 4.7 3.8 3.7 3.8 2.9 3.1 2.8 2.5 2.4 3.1 3.3 3.1 2.3 2.1 2.2 3.8 2.8
## 5 1/5/1998 2.6 2.1 1.6 1.4 0.9 1.5 1.2 1.4 1.3 1.4 2.2   2   3   3 3.1 3.1 2.7
## 6 1/6/1998 3.1 3.5 3.3 2.5 1.6 1.7 1.6 1.6 2.3 1.8 2.5 3.9 3.4 2.7 3.4 2.5 2.2
##   X19 X20 X21 X22 X23 X24 X25 X26 X27  X28  X29  X30  X31  X32  X33  X34  X35
## 1 3.5 3.5 2.9 3.2 3.2 2.8 2.6 5.5 3.1  5.2  6.1  6.1  6.1  6.1  5.6  5.2  5.4
## 2   3 2.6   3 2.2 2.3 2.5 2.8 5.5 3.4 15.1 15.3 15.6 15.6 15.9 16.2 16.2 16.2
## 3 4.4 3.5 2.7 2.9 3.9 4.1 4.6 5.6 3.5 16.6 16.7 16.7 16.8 16.8 16.8 16.9 16.9
## 4 2.4 1.9 3.2 4.1 3.9 4.5 4.3 4.7 3.2 18.3 18.2 18.3 18.4 18.6 18.6 18.5 18.7
## 5   3 2.4 2.8 2.5 2.5 3.7 3.4 3.7 2.3 18.8 18.6 18.5 18.5 18.6 18.9 19.2 19.4
## 6 4.4 4.3 3.2 6.2 6.8 5.1   4 6.8 3.2 18.9 19.5 19.6 19.5 19.5 19.5 19.4 19.2
##    X36  X37  X38  X39  X40  X41  X42  X43  X44  X45  X46  X47  X48  X49  X50
## 1  7.2 10.6 14.5 17.2 18.3 18.9 19.1 18.9 18.3 17.3 16.8 16.1 15.4 14.9 14.8
## 2 16.6 17.8 19.4 20.6 21.2 21.8 22.4 22.1 20.8 19.1 18.1 17.2 16.5 16.1   16
## 3 17.1 17.6 19.1 21.3 21.8   22 22.1 22.2 21.3 19.8 18.6   18   18 18.2 18.3
## 4 18.6 18.8   19   19 19.3 19.4 19.6 19.2 18.9 18.8 18.6 18.5 18.3 18.5 18.8
## 5 19.8 20.5 21.1 21.9 23.8 25.1 25.8   26 25.6 24.2 22.9 21.6   20 19.5 19.1
## 6 19.1 19.5 19.6 18.6 18.6 18.9 19.2 19.3 19.2 18.8 17.6 16.9 15.6 15.4 15.9
##    X51  X52  X53  X54  X55   X56   X57    X58  X59  X60  X61   X62    X63   X64
## 1   15 19.1 12.5  6.7 0.11  3.83  0.14   1612 -2.3  0.3 7.18  0.12 3178.5 -15.5
## 2 16.2 22.4 17.8    9 0.25 -0.41  9.53 1594.5 -2.2 0.96 8.24   7.3   3172 -14.5
## 3 18.4 22.2 18.7    9 0.56  0.89 10.17 1568.5  0.9 0.54  3.8  4.42   3160 -15.9
## 4 18.9 19.6 18.7  9.9 0.89 -0.34  8.58 1546.5    3 0.77 4.17  8.11 3145.5 -16.8
## 5 19.1 26.0 21.1    ?    ?     ?     ?      ?    ?    ?    ?     ?      ?     ?
## 6 15.8 19.6 18.5 14.4 0.68  1.52  8.62 1499.5  4.3 0.61 9.04 10.81   3111 -11.8
##    X65   X66   X67  X68   X69   X70   X71 X72  X73 X74
## 1 0.15 10.67 -1.56 5795 -12.1  17.9 10330 -55 0.00   0
## 2 0.48  8.39  3.84 5805 14.05    29 10275 -55 0.00   0
## 3  0.6  6.94   9.8 5790  17.9  41.3 10235 -40 0.00   0
## 4 0.49  8.73 10.54 5775 31.15  51.7 10195 -40 2.08   0
## 5    ?     ?     ?    ?     ?     ?     ?   ? 0.58   0
## 6 0.09 11.98 11.28 5770 27.95 46.25 10120   ? 5.84   0
  1. Converting the data to numeric data
Assignment_Data <- Assignment_Data %>% mutate_if(is.character,as.numeric)
  1. Reading the TEXT file having list of attribute names of “Assignment_Data”
columnNames <- read_csv("E:/shivani_timeseries/Assignment_columnnames.txt", col_names = FALSE)
columnNames
## # A tibble: 74 x 1
##    X1   
##    <chr>
##  1 Date 
##  2 WSR0 
##  3 WSR1 
##  4 WSR2 
##  5 WSR3 
##  6 WSR4 
##  7 WSR5 
##  8 WSR6 
##  9 WSR7 
## 10 WSR8 
## # ... with 64 more rows
  1. Assigning column names to Assignment data
colnames(Assignment_Data) <- columnNames$X1
names(Assignment_Data)
##  [1] "Date"   "WSR0"   "WSR1"   "WSR2"   "WSR3"   "WSR4"   "WSR5"   "WSR6"  
##  [9] "WSR7"   "WSR8"   "WSR9"   "WSR10"  "WSR11"  "WSR12"  "WSR13"  "WSR14" 
## [17] "WSR15"  "WSR16"  "WSR17"  "WSR18"  "WSR19"  "WSR20"  "WSR21"  "WSR22" 
## [25] "WSR23"  "WSR_PK" "WSR_AV" "T0"     "T1"     "T2"     "T3"     "T4"    
## [33] "T5"     "T6"     "T7"     "T8"     "T9"     "T10"    "T11"    "T12"   
## [41] "T13"    "T14"    "T15"    "T16"    "T17"    "T18"    "T19"    "T20"   
## [49] "T21"    "T22"    "T23"    "T_PK"   "T_AV"   "T85"    "RH85"   "U85"   
## [57] "V85"    "HT85"   "T70"    "RH70"   "U70"    "V70"    "HT70"   "T50"   
## [65] "RH50"   "U50"    "V50"    "HT50"   "KI"     "TT"     "SLP"    "SLP_"  
## [73] "Precp"  "Ozone"

#structure of the Data

head(Assignment_Data)
##   Date WSR0 WSR1 WSR2 WSR3 WSR4 WSR5 WSR6 WSR7 WSR8 WSR9 WSR10 WSR11 WSR12
## 1   NA  0.8  1.8  2.4  2.1  2.0  2.1  1.5  1.7  1.9  2.3   3.7   5.5   5.1
## 2   NA  2.8  3.2  3.3  2.7  3.3  3.2  2.9  2.8  3.1  3.4   4.2   4.5   4.5
## 3   NA  2.9  2.8  2.6  2.1  2.2  2.5  2.5  2.7  2.2  2.5   3.1   4.0   4.4
## 4   NA  4.7  3.8  3.7  3.8  2.9  3.1  2.8  2.5  2.4  3.1   3.3   3.1   2.3
## 5   NA  2.6  2.1  1.6  1.4  0.9  1.5  1.2  1.4  1.3  1.4   2.2   2.0   3.0
## 6   NA  3.1  3.5  3.3  2.5  1.6  1.7  1.6  1.6  2.3  1.8   2.5   3.9   3.4
##   WSR13 WSR14 WSR15 WSR16 WSR17 WSR18 WSR19 WSR20 WSR21 WSR22 WSR23 WSR_PK
## 1   5.4   5.4   4.7   4.3   3.5   3.5   2.9   3.2   3.2   2.8   2.6    5.5
## 2   4.3   5.5   5.1   3.8   3.0   2.6   3.0   2.2   2.3   2.5   2.8    5.5
## 3   4.6   5.6   5.4   5.2   4.4   3.5   2.7   2.9   3.9   4.1   4.6    5.6
## 4   2.1   2.2   3.8   2.8   2.4   1.9   3.2   4.1   3.9   4.5   4.3    4.7
## 5   3.0   3.1   3.1   2.7   3.0   2.4   2.8   2.5   2.5   3.7   3.4    3.7
## 6   2.7   3.4   2.5   2.2   4.4   4.3   3.2   6.2   6.8   5.1   4.0    6.8
##   WSR_AV   T0   T1   T2   T3   T4   T5   T6   T7   T8   T9  T10  T11  T12  T13
## 1    3.1  5.2  6.1  6.1  6.1  6.1  5.6  5.2  5.4  7.2 10.6 14.5 17.2 18.3 18.9
## 2    3.4 15.1 15.3 15.6 15.6 15.9 16.2 16.2 16.2 16.6 17.8 19.4 20.6 21.2 21.8
## 3    3.5 16.6 16.7 16.7 16.8 16.8 16.8 16.9 16.9 17.1 17.6 19.1 21.3 21.8 22.0
## 4    3.2 18.3 18.2 18.3 18.4 18.6 18.6 18.5 18.7 18.6 18.8 19.0 19.0 19.3 19.4
## 5    2.3 18.8 18.6 18.5 18.5 18.6 18.9 19.2 19.4 19.8 20.5 21.1 21.9 23.8 25.1
## 6    3.2 18.9 19.5 19.6 19.5 19.5 19.5 19.4 19.2 19.1 19.5 19.6 18.6 18.6 18.9
##    T14  T15  T16  T17  T18  T19  T20  T21  T22  T23 T_PK T_AV  T85 RH85   U85
## 1 19.1 18.9 18.3 17.3 16.8 16.1 15.4 14.9 14.8 15.0 19.1 12.5  6.7 0.11  3.83
## 2 22.4 22.1 20.8 19.1 18.1 17.2 16.5 16.1 16.0 16.2 22.4 17.8  9.0 0.25 -0.41
## 3 22.1 22.2 21.3 19.8 18.6 18.0 18.0 18.2 18.3 18.4 22.2 18.7  9.0 0.56  0.89
## 4 19.6 19.2 18.9 18.8 18.6 18.5 18.3 18.5 18.8 18.9 19.6 18.7  9.9 0.89 -0.34
## 5 25.8 26.0 25.6 24.2 22.9 21.6 20.0 19.5 19.1 19.1 26.0 21.1   NA   NA    NA
## 6 19.2 19.3 19.2 18.8 17.6 16.9 15.6 15.4 15.9 15.8 19.6 18.5 14.4 0.68  1.52
##     V85   HT85  T70 RH70  U70   V70   HT70   T50 RH50   U50   V50 HT50     KI
## 1  0.14 1612.0 -2.3 0.30 7.18  0.12 3178.5 -15.5 0.15 10.67 -1.56 5795 -12.10
## 2  9.53 1594.5 -2.2 0.96 8.24  7.30 3172.0 -14.5 0.48  8.39  3.84 5805  14.05
## 3 10.17 1568.5  0.9 0.54 3.80  4.42 3160.0 -15.9 0.60  6.94  9.80 5790  17.90
## 4  8.58 1546.5  3.0 0.77 4.17  8.11 3145.5 -16.8 0.49  8.73 10.54 5775  31.15
## 5    NA     NA   NA   NA   NA    NA     NA    NA   NA    NA    NA   NA     NA
## 6  8.62 1499.5  4.3 0.61 9.04 10.81 3111.0 -11.8 0.09 11.98 11.28 5770  27.95
##      TT   SLP SLP_ Precp Ozone
## 1 17.90 10330  -55  0.00     0
## 2 29.00 10275  -55  0.00     0
## 3 41.30 10235  -40  0.00     0
## 4 51.70 10195  -40  2.08     0
## 5    NA    NA   NA  0.58     0
## 6 46.25 10120   NA  5.84     0

Visualising the Missing Data

aggr(Assignment_Data[,-c(1)], combined = TRUE, numbers = TRUE)
## Warning in plot.aggr(res, ...): not enough vertical space to display frequencies
## (too many combinations)

From the detailed summary of each variable we observes that many attributes have missing data.

Below are the attributes for which we will be forecasting:-

WSR_PK: continuous. peek wind speed – resultant (meaning average of wind vector) T_PK: continuous. Peak T T_AV: continuous. Average T T85: continuous. T at 850 hpa level (or about 1500 m height) RH85: continuous. Relative Humidity at 850 hpa HT85: continuous. Geopotential height at 850 hpa, it is about the same as height at low altitude T70: continuous. T at 700 hpa level (roughly 3100 m height) KI: continuous. K-Index TT: continuous. T-Totals SLP: continuous. Sea level pressure SLP_: continuous. SLP change from previous day

Subselecting the variables to be forecasted from the Assignment_Data and storing it as new dataframe as “ozoneData”

ozoneData <- subset(Assignment_Data,select = c("WSR_PK","T_PK","T_AV","T85","RH85","HT85","T70","KI","TT","SLP","SLP_"))
str(ozoneData)
## 'data.frame':    2534 obs. of  11 variables:
##  $ WSR_PK: num  5.5 5.5 5.6 4.7 3.7 6.8 7 6.8 4.1 3.3 ...
##  $ T_PK  : num  19.1 22.4 22.2 19.6 26 19.6 15.8 14.9 17 22.3 ...
##  $ T_AV  : num  12.5 17.8 18.7 18.7 21.1 18.5 9.7 9 10 16.5 ...
##  $ T85   : num  6.7 9 9 9.9 NA 14.4 12.6 3.35 3.9 7.1 ...
##  $ RH85  : num  0.11 0.25 0.56 0.89 NA 0.68 0.98 0.73 0.38 0.23 ...
##  $ HT85  : num  1612 1594 1568 1546 NA ...
##  $ T70   : num  -2.3 -2.2 0.9 3 NA 4.3 4.6 -2.45 -1.95 -3 ...
##  $ KI    : num  -12.1 14.1 17.9 31.1 NA ...
##  $ TT    : num  17.9 29 41.3 51.7 NA ...
##  $ SLP   : num  10330 10275 10235 10195 NA ...
##  $ SLP_  : num  -55 -55 -40 -40 NA NA -80 25 55 60 ...

#Below displays the detailed summary for each variable of “ozoneData” dataset

summary(ozoneData)
##      WSR_PK           T_PK            T_AV            T85       
##  Min.   :0.800   Min.   : 1.70   Min.   : 0.30   Min.   :-7.10  
##  1st Qu.:3.400   1st Qu.:21.20   1st Qu.:16.00   1st Qu.:10.50  
##  Median :4.100   Median :26.60   Median :22.20   Median :14.30  
##  Mean   :4.172   Mean   :25.58   Mean   :20.84   Mean   :13.58  
##  3rd Qu.:4.800   3rd Qu.:31.10   3rd Qu.:26.80   3rd Qu.:17.40  
##  Max.   :9.600   Max.   :41.60   Max.   :33.60   Max.   :24.50  
##  NA's   :273     NA's   :175     NA's   :175     NA's   :99     
##       RH85             HT85           T70               KI         
##  Min.   :0.0100   Min.   :1351   Min.   :-9.900   Min.   :-56.700  
##  1st Qu.:0.3800   1st Qu.:1512   1st Qu.: 3.500   1st Qu.: -3.575  
##  Median :0.6400   Median :1535   Median : 6.800   Median : 14.925  
##  Mean   :0.5773   Mean   :1531   Mean   : 5.931   Mean   : 10.511  
##  3rd Qu.:0.7900   3rd Qu.:1557   3rd Qu.: 8.800   3rd Qu.: 28.350  
##  Max.   :1.0000   Max.   :1642   Max.   :16.200   Max.   : 42.050  
##  NA's   :105      NA's   :95     NA's   :107      NA's   :136      
##        TT              SLP             SLP_        
##  Min.   :-10.10   Min.   : 9975   Min.   :-135.00  
##  1st Qu.: 32.30   1st Qu.:10130   1st Qu.: -20.00  
##  Median : 41.10   Median :10160   Median :   0.00  
##  Mean   : 37.39   Mean   :10164   Mean   :  -0.12  
##  3rd Qu.: 45.10   3rd Qu.:10195   3rd Qu.:  15.00  
##  Max.   : 59.15   Max.   :10350   Max.   : 140.00  
##  NA's   :125      NA's   :95      NA's   :158
##                    WSR_PK         T_PK         T_AV           T85         RH85
## nbr.val      2.261000e+03 2.359000e+03 2.359000e+03  2.435000e+03 2.429000e+03
## nbr.null     0.000000e+00 0.000000e+00 0.000000e+00  0.000000e+00 0.000000e+00
## nbr.na       2.730000e+02 1.750000e+02 1.750000e+02  9.900000e+01 1.050000e+02
## min          8.000000e-01 1.700000e+00 3.000000e-01 -7.100000e+00 1.000000e-02
## max          9.600000e+00 4.160000e+01 3.360000e+01  2.450000e+01 1.000000e+00
## range        8.800000e+00 3.990000e+01 3.330000e+01  3.160000e+01 9.900000e-01
## sum          9.433100e+03 6.033930e+04 4.916270e+04  3.305585e+04 1.402310e+03
## median       4.100000e+00 2.660000e+01 2.220000e+01  1.430000e+01 6.400000e-01
## mean         4.172092e+00 2.557834e+01 2.084048e+01  1.357530e+01 5.773199e-01
## SE.mean      2.468278e-02 1.472505e-01 1.443902e-01  9.877082e-02 5.238647e-03
## CI.mean.0.95 4.840328e-02 2.887539e-01 2.831450e-01  1.936836e-01 1.027268e-02
## var          1.377491e+00 5.114950e+01 4.918171e+01  2.375507e+01 6.666007e-02
## std.dev      1.173666e+00 7.151888e+00 7.012967e+00  4.873917e+00 2.581861e-01
## coef.var     2.813134e-01 2.796072e-01 3.365069e-01  3.590284e-01 4.472150e-01
##                      HT85           T70            KI            TT
## nbr.val      2.439000e+03  2.427000e+03  2398.0000000  2409.0000000
## nbr.null     0.000000e+00  1.000000e+00     1.0000000     1.0000000
## nbr.na       9.500000e+01  1.070000e+02   136.0000000   125.0000000
## min          1.351000e+03 -9.900000e+00   -56.7000000   -10.1000000
## max          1.642000e+03  1.620000e+01    42.0500000    59.1500000
## range        2.910000e+02  2.610000e+01    98.7500000    69.2500000
## sum          3.735315e+06  1.439485e+04 25205.5000000 90068.5000000
## median       1.535000e+03  6.800000e+00    14.9250000    41.1000000
## mean         1.531494e+03  5.931129e+00    10.5110509    37.3883354
## SE.mean      7.430115e-01  7.849298e-02     0.4230834     0.2287960
## CI.mean.0.95 1.456999e+00  1.539202e-01     0.8296471     0.4486574
## var          1.346489e+03  1.495311e+01   429.2409375   126.1053477
## std.dev      3.669454e+01  3.866925e+00    20.7181306    11.2296637
## coef.var     2.395996e-02  6.519711e-01     1.9710808     0.3003521
##                       SLP         SLP_
## nbr.val      2.439000e+03 2376.0000000
## nbr.null     0.000000e+00  202.0000000
## nbr.na       9.500000e+01  158.0000000
## min          9.975000e+03 -135.0000000
## max          1.035000e+04  140.0000000
## range        3.750000e+02  275.0000000
## sum          2.479048e+07 -285.0000000
## median       1.016000e+04    0.0000000
## mean         1.016420e+04   -0.1199495
## SE.mean      1.061440e+00    0.7350363
## CI.mean.0.95 2.081418e+00    1.4413792
## var          2.747911e+03 1283.7013955
## std.dev      5.242052e+01   35.8287789
## coef.var     5.157369e-03 -298.6988723

OBSERVATION:
Each variable has a large amount of missing data. It could also be noticed that all variables are on differnt scales with respect to each other.

EDA throgh Visualisation Method

#Plotting histogram for each variable

ggplot(gather(ozoneData), aes(value))+geom_histogram(bins = 10)+ facet_wrap(~key,scales='free_x') 
## Warning: Removed 1543 rows containing non-finite values (stat_bin).

OBSERVATIONS
1) “SLP”, “SLP_” and “WSR_PK” variables seems to be normally distributed.
2) KI, RH85, T_AV, T_PK, T70 , T85 and TT all these variables are left skewed. Means most of the observations are observed at right side of the graph having larger value compared to left side of the graph.

Plotting boxplots for each variable

OUTLIERS:

boxplot(ozoneData, main = "Boxplot of all variables")

#Boxplot

boxplot(ozoneData[,-c(6,10)], main = "Boxplot of same scale variables")

DEALING WITH OUTLIERS FOR TIME SERIES DATA

An outlier is a point that falls far from the data points. It’s a point that is extreme in some ways. Outliers in a data could be depicted from plotting outliers. The points which lies outside the whisker box are considerd to be outliers.
Here, we see we have many outliers for each variable. However, these datapoint does not seems to be potential outliers as most of the datapoints occurs in bulk.
For time series data, outliers may be errors, or they may simply be unusual. Many time series models are outliers sensitive means they won’t work well if there are extreme outliers in the data.

If the outliers are genuinely errors, then these outliers could be removed easily[1]. However, simply removing outliers could be misleading. This could lead to loss of usefull information which needs to be taken into account while forecasting. Here, we dont notice any extreme outliers for our dataset. So, we are not removing outliers as they dont seem to be potential outliers.

DEALING WITH MISSING VALUES FOR TIME SERIES DATA

#Plot to visualise missing data using “VIM” package,

aggr(ozoneData, numbers = TRUE, prop = c(TRUE, FALSE))

OBSERVATIONs: 1) On total 2090 observations are present in the data without any missing information.
2) Most of the data is missing for WSR_PK comaparitively to other variables.
3) One interesting point could be observed that we have 12 such datapoints which are missing for all variables of “ozoneData”.

There could be four type of Time Series Data:-

  1. No trend or Seasonality (white noise)
  2. Has Trend but no Seasonality
  3. Has Seasonality but No Trend
  4. Has both Seasonality and Trend

Missing Values of Time Series Data is dealt depending on the type of our time series data.
The R functions for ARIMA models, dynamic regression models and NNAR models will also work correctly without causing errors. However, other modelling functions do not handle missing values including ets(), stlf(), and tbats()[1].

Following are the Time-Series Specific Method for dealing with missing values:

  1. Last Observation Carried Forward (LOCF) & Next Observation Carried Backward (NOCB)
    This is a popular statistical approach to analysis of longitudinal repeat measurement data where some follow-up observations may be missing. Longitudinal data tracks the same sample at various points in time. Both methods can introduce bias in analysis and perform poorly when data has a visible trend[2].

  2. Linear Interpolation
    This method is used for such time series data which has trend but no seasonality.

  3. Seasonal Adjustment + Linear Interpolation
    This method works best for those time series data which has both trend and seasonality.

#Converting all the variables of ozoneData to time series data such that it starts from the starting month of the year 1998.

tsWSR_PK <- ts(ozoneData$WSR_PK,start = decimal_date(as.Date("1998-01-01")), frequency = 365)
tsT_PK <- ts(ozoneData$T_PK,start = decimal_date(as.Date("1998-01-01")), frequency = 365)
tsT_AV <- ts(ozoneData$T_AV,start = decimal_date(as.Date("1998-01-01")), frequency = 365)
tsT85 <- ts(ozoneData$T85,start = decimal_date(as.Date("1998-01-01")), frequency = 365)
tsRH85 <- ts(ozoneData$RH85, start = decimal_date(as.Date("1998-01-01")), frequency = 365)
tsHT85 <- ts(ozoneData$HT85, start = decimal_date(as.Date("1998-01-01")), frequency = 365)
tsT70 <- ts(ozoneData$T70, start = decimal_date(as.Date("1998-01-01")), frequency = 365)
tsKI <- ts(ozoneData$KI, start = decimal_date(as.Date("1998-01-01")), frequency = 365)
tsTT <- ts(ozoneData$TT, start = decimal_date(as.Date("1998-01-01")), frequency = 365)
tsSLP <- ts(ozoneData$SLP, start = decimal_date(as.Date("1998-01-01")), frequency = 365)
tsSLP_ <- ts(ozoneData$SLP_,start = decimal_date(as.Date("1998-01-01")), frequency = 365)

FOR 1st VARIABLE: Forecasting on Peak Wind Speed(WSR_PK)

#Checking the attributes of time series

attributes(tsWSR_PK)
## $tsp
## [1] 1998.00 2004.94  365.00
## 
## $class
## [1] "ts"

#Plotting the series

plot(tsWSR_PK, main = "Peak Wind Speed observed for years (WSR_PK)", ylim = c(0,10), col = "red")

From the plot, we observes that there is no trend but seems to have seasonlity.
Most of the data is missing for year 2002 to 2003 and within year 1998 to 1999 approximately.

Checking the statistics and visualising the missing data in the series:

statsNA(tsWSR_PK)
## [1] "Length of time series:"
## [1] 2534
## [1] "-------------------------"
## [1] "Number of Missing Values:"
## [1] 273
## [1] "-------------------------"
## [1] "Percentage of Missing Values:"
## [1] "10.8%"
## [1] "-------------------------"
## [1] "Stats for Bins"
## [1] "  Bin 1 (634 values from 1 to 634) :      20 NAs (3.15%)"
## [1] "  Bin 2 (634 values from 635 to 1268) :      0 NAs (0%)"
## [1] "  Bin 3 (634 values from 1269 to 1902) :      227 NAs (35.8%)"
## [1] "  Bin 4 (632 values from 1903 to 2534) :      26 NAs (4.11%)"
## [1] "-------------------------"
## [1] "Longest NA gap (series of consecutive NAs)"
## [1] "183 in a row"
## [1] "-------------------------"
## [1] "Most frequent gap size (series of consecutive NA series)"
## [1] "183 NA in a row (occuring 1 times)"
## [1] "-------------------------"
## [1] "Gap size accounting for most NAs"
## [1] "183 NA in a row (occuring 1 times, making up for overall 183 NAs)"
## [1] "-------------------------"
## [1] "Overview NA series"
## [1] "  7 NA in a row: 1 times"
## [1] "  10 NA in a row: 1 times"
## [1] "  12 NA in a row: 1 times"
## [1] "  15 NA in a row: 1 times"
## [1] "  20 NA in a row: 1 times"
## [1] "  26 NA in a row: 1 times"
## [1] "  183 NA in a row: 1 times"
plotNA.distribution(tsWSR_PK)

As we see there is a large gap in the missing data, so locf and NOCB would not seems the good technique to implement here.

Also, as no trend is being observed here, so not necessary to use “Kalman filter with arima state space model” as it works best for series having trend and seasonality.

Hence, using “Linear interpolation” technique to impute missing values. This method works good for series having seasonality but no trend type of time series data.

#Impute the missing values with “na_interpolation” function of “imputeTS package” and visualise the imputed values in the time series

tsWSR_PK.imp <- na_interpolation(tsWSR_PK, method = "linear")
plotNA.imputations(tsWSR_PK, tsWSR_PK.imp)

#Decomposition Of Time Series

decomp <- decompose(tsWSR_PK.imp)
plot(decomp, col = "red")

OBSERVATIONS:
1)There does not seems to have any trend. 2)We observe some seasonality in the series. Means there is some seasonality in the movement of wind speed. Seasonality describes cyclical effects due to the time of the year.

Let’s explore this seasonality pattern by plotting the decomp$seasonal components such that y-axis represents Seasonality Index and x-axis has month

plot(decomp$seasonal[1:365], type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")

#las is for numbers to appear vertical

OBSERVATIONS:
1) There is 5 - 10% increase in the speed of wind in starting and end months of a year and we could think the reason because of arrival of winter i.e. seasonal change. 2) There is a huge drop of wind speed for mid-months of the year

#Stationarity check of time series

STATIONARY TIME SERIES: It means when there is no change in the mean, variance or co-variance with time within the series. ARIMA models works with stationary series.
Sometimes it is difficult to interpret the series is stationary or not by just looking. So there are some statistical tests which hepls to identify the stationarity of the series.

We could check the stationarity of time series by ADF and KPSS Tests:

  1. Augmented Dickey- Fuller(ADF) t-statistic test for unit root

The null and alternate hypothesis of this test are:

H0: The series has a unit root(value of a =1)
Ha: The series has no unit root

If we fail to reject the null hypothesis, we can say that the series is non-stationary. This means that the series can be linear or difference stationary[5]

adf.test(tsWSR_PK.imp)
## Warning in adf.test(tsWSR_PK.imp): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  tsWSR_PK.imp
## Dickey-Fuller = -6.5322, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary

NOTE: Here p-value is less, so the series is stationary

  1. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) for level or trend stationarity

KPSS is another test for checking the stationarity of a time series.The null and alternate hypothesis for the KPSS test are opposite that of the ADF test,

H0: The process is trend stationary
Ha: The series has a unit root(series is not stationary)

kpss.test(tsWSR_PK.imp, null="Trend")
## Warning in kpss.test(tsWSR_PK.imp, null = "Trend"): p-value smaller than printed
## p-value
## 
##  KPSS Test for Trend Stationarity
## 
## data:  tsWSR_PK.imp
## KPSS Trend = 0.62045, Truncation lag parameter = 8, p-value = 0.01

NOTE: Here p-value is less, so the series is not stationary.

Following are the possible outcomes:

CASE1: BOTH TESTS RESULTS NOT STATIONARY, THEN SERIES IS NON STATIONARY
CASE2: BOTH TESTS RESULTS STATIONARY, THEN SERIES IS STATIONARY
CASE3: IF KPSS RESULTS STATIONARY AND ADF NOT STATIONARY, MEANS TREND STATIONARY(COULD REMOVE THE TREND TO MAKE IT STATIONARY)
CASE4: IF KPSS RESULTS NOT STATIONARY AND ADF STATIONARY, MEANS DIFFERENCE STATIONARY(COULD USE DIFFERENCING TO MAKE SERIES STATIONARY)

NOTE: From the both tests results, it leads to CASE4, we could recheck by visualising it i.e. by plotting ACF. Second, we can check each for characteristics of stationarity by looking at the autocorrelation functions (ACF) of each signal. For a stationary signal, because we expect no dependence with time, we would expect the ACF to go to 0 for each time lag (τ). Lets visualize the signals and ACFs.

Plotting ACF to check the auto-corelations

acf(tsWSR_PK.imp,lag.max = length(tsWSR_PK.imp),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Here, we observes most of the data lies outside the blue-dotted lines means the significance level. Now, we will take the difference to make our data stationary.

It seems difference of 1 is sufficient to make the series stationary.

tsWSR_PK.sta <- diff(tsWSR_PK.imp)
acf(tsWSR_PK.sta, lag.max = length(tsWSR_PK.sta),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Hence, after differencing our series is now transformed to stationary time series.

#Visualising the stationary series

plot(tsWSR_PK.sta, col = "red")

#CREATING MODELS AND FORECASTING: USING EXPONENTIAL SMOOTHING AND ARIMA TECHNIQUES FOR TIME SERIES FORECASTING

##Exponential Smoothing TECHNIQUE FOR FORECASTING

It is one of the most popular time series technique which uses historical data’s inherent characterstics as input to forecast. The forecasting horizon should be short-term such as the next quarter or potentially six months[5]. There are different exponential smoothing methods that differ from each other in the components of the time series that are modeled[5].

This includes “Simple Exponential Smoothing(ses)”- uses one smoothing constant ,“Holt exponential smoothing(holt)” - uses 2 smoothing constant, “Holt-Winter Exponential Smoothing(hw)”- uses 3 smoothing constant and “Automated exponential smoothing forecasts”.

For “ses” and “Holt exponential smoothing” methods ,the seasonal fluctuations in the data are not part of the model.The method to do so in exponential smoothing is by using “Holt-Winter exponential smoothing”, “Automated exponential smoothing forecasts” methods.

As for “tsWSR_PK.imp” does not have trend but seems to have seasonality, will fit automated exponential smoothening model

##ARIMA TECHNIQUE

ARIMA stands for AUTOREGRESSIVE MOVING AVERAGE MODEL.
a. Auto Regressive (p) - Lags of the variables itself
b. Integrated(d) - Differencing steps required to make stationary
c. Moving Average - Lags of previous information

ARIMA are also renamed as:

AR Model i.e. ARIMA(1,00), MA model i.e. ARIMA(0,0,1) and ARMA model i.e. ARIMA(0,0,1).
ARIMA gives results based on lowest AIC.

#Fit automated exponential smoothing model and ARIMA models to training dataset and calculate forecast accuracy

1.Train - Test Split of the data

train <- window(tsWSR_PK.imp, end = c(2004, 3))
test <- window(tsWSR_PK.imp, start = c(2004, 4))

#Model 1 : Auto Exponential Smoothening Model

set.seed(301)
fit_auto_train <- forecast(train)
summary(fit_auto_train)
## 
## Forecast method: STL +  ETS(A,N,N)
## 
## Model Information:
## ETS(A,N,N) 
## 
## Call:
##  ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend) 
## 
##   Smoothing parameters:
##     alpha = 0.1273 
## 
##   Initial states:
##     l = 4.8892 
## 
##   sigma:  0.8917
## 
##      AIC     AICc      BIC 
## 16371.83 16371.84 16388.91 
## 
## Error measures:
##                        ME      RMSE       MAE       MPE     MAPE      MASE
## Training set -0.004812517 0.8912478 0.6843666 -5.002415 17.82928 0.4926418
##                   ACF1
## Training set 0.2244226
## 
## Forecasts:
##           Point Forecast         Lo 80    Hi 80        Lo 95     Hi 95
## 2004.0082       3.779444  2.6367426562 4.922145  2.031833204  5.527054
## 2004.0110       2.723146  1.5712234928 3.875068  0.961432671  4.484859
## 2004.0137       3.393407  2.2323367772 4.554477  1.617703352  5.169111
## 2004.0164       4.927936  3.7577893676 6.098082  3.138351195  6.717521
## 2004.0192       3.436528  2.2573746637 4.615681  1.633168728  5.239887
## 2004.0219       3.169722  1.9816309000 4.357814  1.352693342  4.986751
## 2004.0247       2.920149  1.7231861814 4.117112  1.089552333  4.750746
## 2004.0274       3.762802  2.5570329718 4.968571  1.918737386  5.606867
## 2004.0301       3.573609  2.3590972927 4.788120  1.716173770  5.431044
## 2004.0329       3.544942  2.3217501589 4.768133  1.674231775  5.415651
## 2004.0356       2.945395  1.7135849988 4.177205  1.061504130  4.829286
## 2004.0384       4.240819  3.0004502531 5.481188  2.343838602  6.137800
## 2004.0411       3.996102  2.7472330098 5.244971  2.086121626  5.906083
## 2004.0438       4.717959  3.4606469679 5.975271  2.795066272  6.640851
## 2004.0466       4.737302  3.4716033233 6.003000  2.801583126  6.673020
## 2004.0493       3.759253  2.4852232698 5.033282  1.810792795  5.707713
## 2004.0521       3.096605  1.8142984563 4.378912  1.135486356  5.057724
## 2004.0548       3.350830  2.0602994403 4.641361  1.377133817  5.324526
## 2004.0575       3.066036  1.7673336063 4.364739  1.079842028  5.052230
## 2004.0603       3.102256  1.7954326157 4.409079  1.103642134  5.100870
## 2004.0630       3.430585  2.1156913098 4.745479  1.419628473  5.441542
## 2004.0658       3.810230  2.4873150531 5.133146  1.787005926  5.833455
## 2004.0685       3.684722  2.3538339018 5.015611  1.649304076  5.720141
## 2004.0712       3.767927  2.4291132473 5.106742  1.720387859  5.815467
## 2004.0740       4.542769  3.1960757464 5.889462  2.483179486  6.602358
## 2004.0767       4.044270  2.6897436837 5.398796  1.972700812  6.115839
## 2004.0795       3.166627  1.8043131668 4.528942  1.083147527  5.250107
## 2004.0822       3.370927  2.0008693368 4.740986  1.275604363  5.466251
## 2004.0849       3.333514  1.9557553009 4.711272  1.226414034  5.440613
## 2004.0877       3.979715  2.5942989533 5.365131  1.860904050  6.098526
## 2004.0904       3.983918  2.5908863977 5.376949  1.853460140  6.114375
## 2004.0932       3.974583  2.5739777451 5.375188  1.832542052  6.116624
## 2004.0959       3.734267  2.3261287365 5.142406  1.580705173  5.887830
## 2004.0986       4.358673  2.9430416558 5.774305  2.193651444  6.523695
## 2004.1014       3.700552  2.2774668011 5.123638  1.524130826  5.876974
## 2004.1041       3.025586  1.5950860172 4.456087  0.837824838  5.213348
## 2004.1068       3.602042  2.1641648744 5.039919  1.402998733  5.801085
## 2004.1096       4.244309  2.7990929321 5.689525  2.034041760  6.454576
## 2004.1123       3.860317  2.4077993346 5.312835  1.638882761  6.081752
## 2004.1151       4.466015  3.0062312219 5.925798  2.233468581  6.698561
## 2004.1178       3.873607  2.4065937986 5.340619  1.630004139  6.117209
## 2004.1205       3.712159  2.2379520352 5.186366  1.457554123  5.966763
## 2004.1233       3.670666  2.1893003366 5.152032  1.405112666  5.936220
## 2004.1260       3.747240  2.2587494555 5.235730  1.470790254  6.023689
## 2004.1288       3.922939  2.4273578199 5.418520  1.635645053  6.210233
## 2004.1315       4.119609  2.6169703520 5.622247  1.821521732  6.417695
## 2004.1342       4.632859  3.1231964053 6.142521  2.324029397  6.941688
## 2004.1370       3.196043  1.6793889779 4.712697  0.876520801  5.515565
## 2004.1397       3.585999  2.0623855278 5.109613  1.255833167  5.916165
## 2004.1425       3.970228  2.4396867329 5.500770  1.629466940  6.310990
## 2004.1452       3.526284  1.9888461093 5.063723  1.174975411  5.877593
## 2004.1479       4.046393  2.5020883043 5.590697  1.684583005  6.408202
## 2004.1507       5.512837  3.9616976546 7.063977  3.140573842  7.885101
## 2004.1534       5.017057  3.4591116276 6.575002  2.634385177  7.399729
## 2004.1562       4.638323  3.0736012619 6.203044  2.245287844  7.031357
## 2004.1589       4.060573  2.4891047720 5.632041  1.657219852  6.463926
## 2004.1616       3.067857  1.4896712230 4.646043  0.654230069  5.481484
## 2004.1644       4.305421  2.7205457433 5.890296  1.881563429  6.729279
## 2004.1671       4.317462  2.7259256421 5.908999  1.883417052  6.751507
## 2004.1699       3.229872  1.6317019472 4.828042  0.785681779  5.674062
## 2004.1726       3.596775  1.9919983829 5.201551  1.142481151  6.051068
## 2004.1753       3.603754  1.9923989939 5.215110  1.139399036  6.068110
## 2004.1781       5.203353  3.5854455322 6.821261  2.728977010  7.677729
## 2004.1808       5.742314  4.1178801074 7.366747  3.257957011  8.226670
## 2004.1836       4.077321  2.4463875541 5.708254  1.583023707  6.571618
## 2004.1863       4.062589  2.4251821780 5.699996  1.558391238  6.566787
## 2004.1890       4.054137  2.4102815684 5.697993  1.540077032  6.568197
## 2004.1918       4.211363  2.5610845647 5.861642  1.687479770  6.735247
## 2004.1945       4.080703  2.4240262215 5.737380  1.547034353  6.614372
## 2004.1973       3.322772  1.6597209188 4.985823  0.779355007  5.866188
## 2004.2000       3.588025  1.9186251706 5.257426  1.034898097  6.141153
## 2004.2027       3.397057  1.7213313133 5.072782  0.834255814  5.959858
## 2004.2055       3.254734  1.5727069336 4.936761  0.682295600  5.827172
## 2004.2082       3.999101  2.3107956301 5.687406  1.417060913  6.581140
## 2004.2110       3.685882  1.9913226011 5.380442  1.094276813  6.277488
## 2004.2137       3.094809  1.3940176465 4.795601  0.493672964  5.695946
## 2004.2164       2.764765  1.0577647258 4.471766  0.154133192  5.375397
## 2004.2192       4.044633  2.3314458499 5.757820  1.424539377  6.664727
## 2004.2219       3.374334  1.6549827941 5.093686  0.744813166  6.003855
## 2004.2247       3.551319  1.8258256667 5.276813  0.912404541  6.190234
## 2004.2274       4.407270  2.6756559579 6.138884  1.758994868  7.055545
## 2004.2301       4.319268  2.5815546970 6.056980  1.661665054  6.976870
## 2004.2329       3.893169  2.1493784254 5.636959  1.226271521  6.560066
## 2004.2356       3.702789  1.9529418731 5.452636  1.026628882  6.378949
## 2004.2384       4.461754  2.7058711994 6.217636  1.776363179  7.147144
## 2004.2411       4.059629  2.2977320729 5.821527  1.365039969  6.754219
## 2004.2438       3.713391  1.9454996713 5.481283  1.009634316  6.417149
## 2004.2466       2.779052  1.0051857139 4.552918  0.066157832  5.491945
## 2004.2493       3.345216  1.5653961932 5.125036  0.623216399  6.067216
## 2004.2521       3.849864  2.0641096315 5.635618  1.118788434  6.580939
## 2004.2548       3.477930  1.6862608504 5.269599  0.737808654  6.218051
## 2004.2575       3.254087  1.4565231501 5.051651  0.504950258  6.003224
## 2004.2603       3.691276  1.8878359855 5.494716  0.933152598  6.449399
## 2004.2630       4.037317  2.2280206269 5.846614  1.270236845  6.804398
## 2004.2658       4.357618  2.5424838605 6.172753  1.581609689  7.133627
## 2004.2685       3.964785  2.1438309881 5.785738  1.179876334  6.749693
## 2004.2712       3.900952  2.0741979499 5.727707  1.107172626  6.694732
## 2004.2740       3.783326  1.9507891987 5.615862  0.980702925  6.585949
## 2004.2767       3.364896  1.5265949780 5.203196  0.553457383  6.176334
## 2004.2795       3.560533  1.7164864364 5.404580  0.740307057  6.380759
## 2004.2822       3.535205  1.6854301510 5.384980  0.706218437  6.364192
## 2004.2849       3.996866  2.1413806336 5.852352  1.159145945  6.834586
## 2004.2877       3.584990  1.7238114578 5.446168  0.738563071  6.431417
## 2004.2904       4.173718  2.3068638917 6.040572  1.318610996  7.028825
## 2004.2932       4.432122  2.5596095426 6.304635  1.568361245  7.295883
## 2004.2959       3.863826  1.9856723948 5.741980  0.991437720  6.736215
## 2004.2986       3.575377  1.6915982567 5.459155  0.694386148  6.456367
## 2004.3014       4.441213  2.5518263600 6.330599  1.551645681  7.330780
## 2004.3041       4.285115  2.3901370821 6.180092  1.386996617  7.183232
## 2004.3068       4.075140  2.1745883000 5.975693  1.168496757  6.981784
## 2004.3096       3.870534  1.9644231496 5.776644  0.955389159  6.785678
## 2004.3123       3.943879  2.0322262913 5.855532  1.020258408  6.867500
## 2004.3151       3.751594  1.8344149347 5.668773  0.819521641  6.683666
## 2004.3178       2.834084  0.9113941378 4.756773 -0.106416159  5.774583
## 2004.3205       3.534298  1.6061140713 5.462482  0.585395109  6.483201
## 2004.3233       3.867054  1.9333912999 5.800717  0.909771936  6.824337
## 2004.3260       3.804647  1.8655205422 5.743774  0.839008972  6.770285
## 2004.3288       2.973231  1.0286564533 4.917806 -0.000739197  5.947201
## 2004.3315       2.796357  0.8463489544 4.746364 -0.185922718  5.778636
## 2004.3342       3.458482  1.5030561937 5.413907  0.467916489  6.449047
## 2004.3370       3.539810  1.5789813686 5.500638  0.540981557  6.538638
## 2004.3397       4.363802  2.3975861060 6.330019  1.356734046  7.370871
## 2004.3425       3.758230  1.7866401390 5.729819  0.742943626  6.773516
## 2004.3452       3.127244  1.1502955339 5.104192  0.103762298  6.150725
## 2004.3479       2.907374  0.9250815456 4.889667 -0.124280744  5.939029
## 2004.3507       3.905534  1.9179116375 5.893156  0.865727901  6.945340
## 2004.3534       3.628857  1.6359193778 5.621795  0.580921739  6.676793
## 2004.3562       3.599975  1.6017358440 5.598215  0.543931789  6.656019
## 2004.3589       3.759797  1.7562703617 5.763324  0.695667316  6.823927
## 2004.3616       3.254322  1.2455216705 5.263122  0.182127002  6.326517
## 2004.3644       3.533048  1.5189882523 5.547108  0.452809270  6.613287
## 2004.3671       3.389051  1.3697453207 5.408357  0.300789277  6.477313
## 2004.3699       3.453453  1.4289142415 5.477991  0.357188332  6.549717
## 2004.3726       2.461124  0.4313663949 4.490881 -0.643122240  5.565370
## 2004.3753       3.253690  1.2187270821 5.288653  0.141482807  6.365897
## 2004.3781       3.161180  1.1210245026 5.201335  0.041031619  6.281328
## 2004.3808       2.947613  0.9022788353 4.992947 -0.180455680  6.075682
## 2004.3836       3.285723  1.2352229798 5.336223  0.149753758  6.421693
## 2004.3863       3.622941  1.5672876108 5.678594  0.479090554  6.766791
## 2004.3890       3.678382  1.6175883664 5.739175  0.526670297  6.830093
## 2004.3918       3.070614  1.0046929152 5.136534 -0.088939398  6.230166
## 2004.3945       3.293574  1.2225390411 5.364610  0.126199204  6.460949
## 2004.3973       3.300423  1.2242857037 5.376560  0.125245013  6.475601
## 2004.4000       3.755199  1.6739720153 5.836426  0.572237092  6.938161
## 2004.4027       3.529622  1.4433180204 5.615926  0.338895438  6.720348
## 2004.4055       2.987846  0.8964772658 5.079215 -0.210626452  6.186318
## 2004.4082       2.847565  0.7511432617 4.943986 -0.358635114  6.053764
## 2004.4110       2.973952  0.8724899323 5.075413 -0.239956671  6.187860
## 2004.4137       3.396945  1.2904550652 5.503435  0.175346619  6.618544
## 2004.4164       3.526150  1.4146438609 5.637657  0.296879911  6.755421
## 2004.4192       3.086001  0.9694900075 5.202512 -0.150923153  6.322925
## 2004.4219       3.110623  0.9891195415 5.232127 -0.133936580  6.355183
## 2004.4247       2.457786  0.3313012370 4.584270 -0.794391640  5.709963
## 2004.4274       2.625129  0.4936750222 4.756583 -0.634648448  5.884906
## 2004.4301       2.943813  0.8074012442 5.080224 -0.323546701  6.211172
## 2004.4329       2.469123  0.3277651604 4.610481 -0.805801184  5.744047
## 2004.4356       2.449614  0.3033214405 4.595907 -0.832857268  5.732085
## 2004.4384       2.973090  0.8218738272 5.124306 -0.316911253  6.263091
## 2004.4411       2.703488  0.5473592737 4.859616 -0.594026227  6.001002
## 2004.4438       2.572985  0.4119551201 4.734014 -0.732024889  5.877994
## 2004.4466       2.819954  0.6540347560 4.985874 -0.492533892  6.132443
## 2004.4493       2.973583  0.8027844426 5.144382 -0.346367012  6.293533
## 2004.4521       3.077371  0.9017037201 5.253037 -0.250024749  6.404766
## 2004.4548       2.912194  0.7316698747 5.092718 -0.422629856  6.247018
## 2004.4575       3.009123  0.8237527349 5.194494 -0.333112542  6.351359
## 2004.4603       3.075898  0.8856919595 5.266104 -0.273733187  6.425530
## 2004.4630       3.384883  1.1898513985 5.579914  0.027872022  6.741893
## 2004.4658       3.143759  0.9439135509 5.343605 -0.220614453  6.508133
## 2004.4685       2.772826  0.5681760331 4.977475 -0.598895033  6.144547
## 2004.4712       3.111874  0.9024311661 5.321318 -0.267177433  6.490926
## 2004.4740       3.144519  0.9302927683 5.358746 -0.241847870  6.530886
## 2004.4767       3.527727  1.3087273093 5.746726  0.134060089  6.921393
## 2004.4795       3.293097  1.0693353337 5.516859 -0.107853045  6.694047
## 2004.4822       2.743152  0.5146374653 4.971666 -0.665066685  6.151370
## 2004.4849       2.983511  0.7502546910 5.216768 -0.431959876  6.398982
## 2004.4877       3.031971  0.7939824901 5.269960 -0.390737175  6.454680
## 2004.4904       3.146489  0.9037776685 5.389200 -0.283441808  6.576419
## 2004.4932       2.952027  0.7046033063 5.199450 -0.485110729  6.389164
## 2004.4959       3.394896  1.1427699759 5.647022 -0.049433399  6.839225
## 2004.4986       3.195302  0.9384833518 5.452120 -0.256204176  6.646808
## 2004.5014       3.141474  0.8799723573 5.402975 -0.317194169  6.600142
## 2004.5041       2.923191  0.6570166136 5.189366 -0.542623788  6.389006
## 2004.5068       3.155422  0.8845838318 5.426260 -0.317525353  6.628370
## 2004.5096       3.016617  0.7411243823 5.292109 -0.463448527  6.496682
## 2004.5123       2.792930  0.5127932231 5.073067 -0.694238382  6.280099
## 2004.5151       3.156801  0.8720286096 5.441573 -0.337456693  6.651058
## 2004.5178       3.072207  0.7828094760 5.361605 -0.429124556  6.573539
## 2004.5205       3.031100  0.7370853644 5.325114 -0.477292460  6.539492
## 2004.5233       3.339231  1.0406096226 5.637853 -0.176207086  6.854669
## 2004.5260       3.010581  0.7073611526 5.313800 -0.511889561  6.533051
## 2004.5288       3.358299  1.0504908914 5.666107 -0.171188978  6.887787
## 2004.5315       2.798293  0.4859055799 5.110681 -0.738198626  6.334786
## 2004.5342       3.084867  0.7679086575 5.401826 -0.458615092  6.628349
## 2004.5370       3.279240  0.9577198623 5.600760 -0.271218667  6.829699
## 2004.5397       3.505831  1.1797585684 5.831904 -0.051590005  7.063253
## 2004.5425       3.148788  0.8181713964 5.479405 -0.415582514  6.713159
## 2004.5452       3.648061  1.3129093643 5.983212  0.076754798  7.219367
## 2004.5479       3.047522  0.7078439413 5.387199 -0.530706628  6.625750
## 2004.5507       3.415722  1.0715271553 5.759917 -0.169414791  7.000859
## 2004.5534       3.036535  0.6878315540 5.385239 -0.555497170  6.628568
## 2004.5562       2.952802  0.5995979728 5.306006 -0.646112956  6.551717
## 2004.5589       3.228395  0.8706997740 5.586091 -0.377388812  6.834179
## 2004.5616       3.511464  1.1492860143 5.873643 -0.101175709  7.124105
## 2004.5644       3.180136  0.8134827281 5.546789 -0.439347636  6.799619
## 2004.5671       2.898319  0.5272005332 5.269438 -0.727994003  6.524633
## 2004.5699       3.356230  0.9806537295 5.731807 -0.276900534  6.989361
## 2004.5726       3.154182  0.7741557806 5.534207 -0.485753790  6.794117
## 2004.5753       3.758840  1.3743732732 6.143307  0.112112790  7.405567
## 2004.5781       3.228186  0.8392868664 5.617086 -0.425320159  6.881693
## 2004.5808       3.349316  0.9559914660 5.742640 -0.310957756  7.009589
## 2004.5836       3.264179  0.8664382604 5.661919 -0.402848835  6.931206
## 2004.5863       3.044852  0.6427029905 5.447000 -0.628917681  6.718621
## 2004.5890       3.034840  0.6282909517 5.441389 -0.645659022  6.715338
## 2004.5918       3.281519  0.8705781386 5.692460 -0.405696885  6.968735
## 2004.5945       2.987035  0.5717095653 5.402360 -0.706886281  6.680956
## 2004.5973       3.097615  0.6779141966 5.517317 -0.602998267  6.798229
## 2004.6000       3.784652  1.3605821573 6.208721  0.077357259  7.491946
## 2004.6027       4.038829  1.6103986285 6.467259  0.324865455  7.752792
## 2004.6055       3.444853  1.0120701738 5.877635 -0.275767138  7.165473
## 2004.6082       2.776498  0.3393707351 5.213626 -0.950766600  6.503763
## 2004.6110       2.948794  0.5073291909 5.390258 -0.785104074  6.682692
## 2004.6137       3.579169  1.1333748328 6.024963 -0.161350291  7.319688
## 2004.6164       3.220736  0.7706198204 5.670851 -0.526393112  6.967864
## 2004.6192       3.293723  0.8392934278 5.748153 -0.460003286  7.047450
## 2004.6219       3.004188  0.5454518139 5.462925 -0.756124673  6.764501
## 2004.6247       2.895269  0.4322332359 5.358304 -0.871619038  6.662157
## 2004.6274       2.825361  0.3580340285 5.292688 -0.948090067  6.598812
## 2004.6301       2.892591  0.4209797326 5.364202 -0.887412240  6.672594
## 2004.6329       3.508446  1.0325583677 5.984334 -0.278097558  7.294990
## 2004.6356       3.325353  0.8451952605 5.805510 -0.467720715  7.118426
## 2004.6384       3.523310  1.0388905324 6.007729 -0.276281608  7.322901
## 2004.6411       2.868934  0.3802603050 5.357608 -0.937164137  6.675033
## 2004.6438       3.259458  0.7665365654 5.752379 -0.553136334  7.072052
## 2004.6466       3.031638  0.5344760691 5.528799 -0.787441463  6.850717
## 2004.6493       3.793565  1.2921700849 6.294959 -0.031988276  7.619118
## 2004.6521       3.725934  1.2203136542 6.231555 -0.106081749  7.557950
## 2004.6548       3.205093  0.6952541312 5.714933 -0.633374548  7.043561
## 2004.6575       2.863936  0.3498850472 5.377987 -0.980973160  6.708845
## 2004.6603       3.304459  0.7862033912 5.822715 -0.546880616  7.155799
## 2004.6630       3.155843  0.6333893566 5.678296 -0.701916740  7.013602
## 2004.6658       2.840049  0.3134052296 5.366693 -1.024119265  6.704218
## 2004.6685       2.707848  0.1770203929 5.238676 -1.162718826  6.578415
## 2004.6712       2.795132  0.2601271291 5.330136 -1.081823159  6.672086
## 2004.6740       3.059165  0.5199909848 5.598340 -0.824166735  6.942497
## 2004.6767       3.324944  0.7816068707 5.868282 -0.564754662  7.214643
## 2004.6795       3.290368  0.7428738288 5.837861 -0.605687916  7.186423
## 2004.6822       3.332360  0.7807163302 5.884003 -0.570042042  7.234761
## 2004.6849       3.567930  1.0121439576 6.123716 -0.340807476  7.476668
## 2004.6877       3.691913  1.1319909281 6.251835 -0.223150018  7.606976
## 2004.6904       3.241411  0.6773589460 5.805462 -0.679967980  7.162789
## 2004.6932       3.113256  0.5450816813 5.681430 -0.814427710  7.040940
## 2004.6959       3.496012  0.9237209994 6.068302 -0.437967360  7.429990
## 2004.6986       3.445482  0.8690818196 6.021882 -0.494782026  7.385746
## 2004.7014       3.180317  0.5998143100 5.760821 -0.766221557  7.126857
## 2004.7041       2.915929  0.3313289386 5.500528 -1.036875502  6.868733
## 2004.7068       2.697170  0.1084802211 5.285860 -1.261889361  6.656229
## 2004.7096       3.285659  0.6928852788 5.878432 -0.679646030  7.250963
## 2004.7123       2.625101  0.0282504506 5.221952 -1.346439186  6.596641
## 2004.7151       2.654321  0.0533995975 5.255242 -1.323444982  6.632087
## 2004.7178       3.179420  0.5744341220 5.784406 -0.804562034  7.163402
## 2004.7205       3.334191  0.7251466940 5.943234 -0.655997687  7.324379
## 2004.7233       3.125878  0.5127826834 5.738974 -0.870506585  7.122263
## 2004.7260       3.821928  1.2047870054 6.439069 -0.180643831  7.824500
## 2004.7288       3.536404  0.9152232503 6.157584 -0.472345848  7.545153
## 2004.7315       3.239520  0.6143067982 5.864734 -0.775397273  7.254438
## 2004.7342       3.905553  1.2763123173 6.534793 -0.115523451  7.926629
## 2004.7370       3.479877  0.8466158018 6.113138 -0.547348404  7.507102
## 2004.7397       3.201934  0.5646580433 5.839209 -0.831431355  7.235299
## 2004.7425       3.228888  0.5876042268 5.870172 -0.810607134  7.268384
## 2004.7452       3.152221  0.5069346958 5.797508 -0.893395412  7.197838
## 2004.7479       3.900346  1.2510634652 6.549629 -0.151382189  7.952075
## 2004.7507       2.944691  0.2914182425 5.597965 -1.113139772  7.002523
## 2004.7534       3.673922  1.0166642118 6.331179 -0.390002990  7.737847
## 2004.7562       3.847909  1.1866725359 6.509145 -0.222100696  7.917918
## 2004.7589       3.778530  1.1133214704 6.443738 -0.297554647  7.854614
## 2004.7616       3.115121  0.4459457552 5.784296 -0.967030119  7.197272
## 2004.7644       3.357914  0.6847779628 6.031049 -0.730294552  7.446122
## 2004.7671       3.465230  0.7881391945 6.142320 -0.629026859  7.559486
## 2004.7699       3.754470  1.0734310867 6.435510 -0.345825417  7.854766
## 2004.7726       3.840091  1.1551088022 6.525074 -0.266235078  7.946418
## 2004.7753       3.983698  1.2947778317 6.672618 -0.128650364  8.096046
## 2004.7781       3.932843  1.2399918318 6.625695 -0.185517631  8.051204
## 2004.7808       3.782285  1.0855072066 6.479062 -0.342080489  7.906650
## 2004.7836       3.325113  0.6244155310 6.025811 -0.805247377  7.455473
## 2004.7863       3.301687  0.5970749860 6.006299 -0.834660127  7.438034
## 2004.7890       3.602056  0.8935351018 6.310577 -0.540269220  7.744381
## 2004.7918       3.280751  0.5683268458 5.993175 -0.867543704  7.429045
## 2004.7945       3.376233  0.6599113569 6.092555 -0.778022451  7.530488
## 2004.7973       3.557491  0.8372772956 6.277705 -0.602716815  7.717699
## 2004.8000       4.054772  1.3306723675 6.778873 -0.111379101  8.220924
## 2004.8027       3.839486  1.1115048373 6.567467 -0.332601059  8.011573
## 2004.8055       3.100940  0.3690832557 5.832796 -1.077074150  7.278953
## 2004.8082       3.573701  0.8379752093 6.309428 -0.610230799  7.757634
## 2004.8110       3.822717  1.0831267023 6.562308 -0.367125015  8.012560
## 2004.8137       3.463569  0.7201188224 6.207018 -0.732175723  7.659313
## 2004.8164       3.313529  0.5662260416 6.060833 -0.888108462  7.515167
## 2004.8192       3.821311  1.0701599748 6.572463 -0.386211629  8.028834
## 2004.8219       3.742743  0.9877485046 6.497737 -0.470657355  7.956143
## 2004.8247       2.114504 -0.6443278758 4.873336 -2.104765157  6.333773
## 2004.8274       2.346998 -0.4156655234 5.109662 -1.878131404  6.572128
## 2004.8301       3.099915  0.3334241487 5.866405 -1.131067522  7.330897
## 2004.8329       3.050182  0.2798695437 5.820494 -1.186645119  7.287008
## 2004.8356       2.909701  0.1355722291 5.683829 -1.332962638  7.152364
## 2004.8384       3.267784  0.4898447090 6.045724 -0.980707588  7.516276
## 2004.8411       3.446398  0.6646531523 6.228144 -0.807913810  7.700711
## 2004.8438       3.057415  0.2718689195 5.842961 -1.202709956  7.317539
## 2004.8466       3.523398  0.7340571754 6.312740 -0.742530872  7.789328
## 2004.8493       3.797508  1.0043767907 6.590640 -0.474217699  8.069234
## 2004.8521       2.847015  0.0500987063 5.643932 -1.430499506  7.124530
## 2004.8548       2.991343  0.1906466606 5.792040 -1.291952566  7.274639
## 2004.8575       3.716269  0.9117972741 6.520740 -0.572800270  8.005338
## 2004.8603       3.934839  1.1265980524 6.743081 -0.359995123  8.229674
## 2004.8630       4.406596  1.5945897378 7.218602  0.106003606  8.707188
## 2004.8658       3.526536  0.7107702500 6.342302 -0.779806173  7.832878
## 2004.8685       3.837702  1.0181817500 6.657223 -0.474382310  8.149787
## 2004.8712       3.596493  0.7732226729 6.419763 -0.721326381  7.914312
## 2004.8740       3.857116  1.0301005999 6.684131 -0.466430815  8.180662
## 2004.8767       3.906089  1.0753337886 6.736844 -0.423177365  8.235355
## 2004.8795       3.270415  0.4359252556 6.104905 -1.064563025  7.605393
## 2004.8822       3.333994  0.4957740387 6.172214 -1.006688766  7.674676
## 2004.8849       4.122678  1.2807326866 6.964622 -0.223702052  8.469057
## 2004.8877       3.804409  0.9587441315 6.650074 -0.547659959  8.156478
## 2004.8904       3.789673  0.9402920990 6.639053 -0.568078772  8.147424
## 2004.8932       3.714606  0.8615146206 6.567696 -0.648820470  8.078032
## 2004.8959       3.560560  0.7037633421 6.417357 -0.808533417  7.929653
## 2004.8986       3.547938  0.6874401529 6.408435 -0.826815733  7.922691
## 2004.9014       3.972143  1.1079492331 6.836336 -0.408263249  8.352549
## 2004.9041       3.755462  0.8875766849 6.623346 -0.630589871  8.141513
## 2004.9068       3.448248  0.5766761459 6.319819 -0.943441972  7.839937
## 2004.9096       4.557044  1.6817906266 7.432297  0.159723449  8.954364
## 2004.9123       5.271880  2.3929499772 8.150811  0.868936232  9.674825
## 2004.9151       5.449443  2.5668396621 8.332046  1.040881833  9.858003
## 2004.9178       3.386854  0.5005829314 6.273124 -1.027316508  7.801024
## 2004.9205       4.479393  1.5894595555 7.369327  0.059620969  8.899166
## 2004.9233       4.873154  1.9795617551 7.766747  0.447786478  9.298522
## 2004.9260       4.640047  1.7428006941 7.537293  0.209091171  9.071003
## 2004.9288       4.375590  1.4746941990 7.276485 -0.060947134  8.812127
## 2004.9315       5.339619  2.4350786024 8.244159  0.897507887  9.781730
## 2004.9342       3.647156  0.7389759607 6.555337 -0.800521719  8.094834
## 2004.9370       3.993534  1.0817176323 6.905349 -0.459704603  8.446772
## 2004.9397       3.401641  0.4861940319 6.317088 -1.057150359  7.860432
## 2004.9425       3.873255  0.9541818322 6.792329 -0.591082323  8.337593
## 2004.9452       3.798746  0.8760506721 6.721442 -0.671130866  8.268623
## 2004.9479       4.834415  1.9081018351 7.760728  0.359005288  9.309824
## 2004.9507       4.485037  1.5551104645 7.414963  0.004101272  8.965972
## 2004.9534       4.628231  1.6946967644 7.561766  0.141777283  9.114686
## 2004.9562       3.974883  1.0377441651 6.912022 -0.517083259  8.466849
## 2004.9589       3.930952  0.9902133642 6.871691 -0.566519663  8.428424
## 2004.9616       4.343130  1.3987960029 7.287464 -0.159840298  8.846100
## 2004.9644       3.899447  0.9515223998 6.847372 -0.609014854  8.407910
## 2004.9671       4.009766  1.0582541754 6.961277 -0.504181718  8.523713
## 2004.9699       4.319703  1.3646093353 7.274797 -0.199722894  8.839129
## 2004.9726       3.333536  0.3748644480 6.292208 -1.191361821  7.858434
## 2004.9753       4.017467  1.0552211304 6.979712 -0.512896890  8.547830
## 2004.9781       4.172191  1.2063765619 7.138006 -0.363630931  8.708014
## 2004.9808       4.346757  1.3773775461 7.316137 -0.194517148  8.888032
## 2004.9836       4.706885  1.7339449938 7.679826  0.160165362  9.253606
## 2004.9863       4.760269  1.7837723662 7.736766  0.208110051  9.312429
## 2004.9890       4.496551  1.5165017390 7.476600 -0.061041013  9.054143
## 2004.9918       4.115070  1.1314726990 7.098667 -0.447948250  8.678088
## 2004.9945       3.541175  0.5540336589 6.528316 -1.027263257  8.109613
## 2004.9973       3.419677  0.4289965089 6.410358 -1.154174151  7.993528
## 2005.0000       3.636529  0.6423130922 6.630745 -0.942729096  8.215787
## 2005.0027       4.057684  1.0599370890 7.055431 -0.526974421  8.642343
## 2005.0055       4.086440  1.0851654886 7.087714 -0.503613143  8.676493
## 2005.0082       3.779444  0.7746465962 6.784241 -0.815996966  8.374885
## 2005.0110       2.723146 -0.2851702519 5.731462 -1.877676561  7.323968
## 2005.0137       3.393407  0.3815762278 6.405238 -1.212790651  7.999605
## 2005.0164       4.927936  1.9125945966 7.943277  0.316369316  9.539503
## 2005.0192       3.436528  0.4176798913 6.455376 -1.180401630  8.053457
## 2005.0219       3.169722  0.1473719210 6.192073 -1.452563687  7.792008
## 2005.0247       2.920149 -0.1056996939 5.945998 -1.707487243  7.547785
## 2005.0274       3.762802  0.7334589694 6.792145 -0.870178381  8.395782
## 2005.0301       3.573609  0.5407753384 6.606442 -1.064709683  8.211927
## 2005.0329       3.544942  0.5086217832 6.581261 -1.098708785  8.188592
## 2005.0356       2.945395 -0.0944069611 5.985197 -1.703580959  7.594371
## 2005.0384       4.240819  1.1975388077 7.284100 -0.413476511  8.895115
## 2005.0411       3.996102  0.9493473951 7.042857 -0.663507142  8.655711
## 2005.0438       4.717959  1.6677336762 7.768184  0.053042015  9.382876
## 2005.0466       4.737302  1.6836099833 7.790993  0.067083286  9.407520
## 2005.0493       3.759253  0.7020986089 6.816407 -0.916261043  8.434767
## 2005.0521       3.096605  0.0359922645 6.157218 -1.584198269  7.777408
## 2005.0548       3.350830  0.2867625358 6.414898 -1.335256812  8.036917
## 2005.0575       3.066036 -0.0014821972 6.133554 -1.625328301  7.757401
## 2005.0603       3.102256  0.0312906905 6.173221 -1.594380115  7.798892
## 2005.0630       3.430585  0.3561769740 6.504994 -1.271316488  8.132487
## 2005.0658       3.810230  0.7323829229 6.888078 -0.896931157  8.517392
## 2005.0685       3.684722  0.6034394706 6.766005 -1.027693195  8.397138
## 2005.0712       3.767927  0.6832128594 6.852642 -0.949736367  8.485591
## 2005.0740       4.542769  1.4546265716 7.630911 -0.180137196  9.265675
## 2005.0767       4.044270  0.9527036934 7.135836 -0.683872604  8.772412
## 2005.0795       3.166627  0.0716411100 6.261614 -1.566745712  7.900001
## 2005.0822       3.370927  0.2725247180 6.469330 -1.367670630  8.109526
## 2005.0849       3.333514  0.2316983588 6.435329 -1.410303523  8.077331
## 2005.0877       3.979715  0.8744906399 7.084939 -0.769315791  8.728746
## 2005.0904       3.983918  0.8752883584 7.092547 -0.770320643  8.738156
## 2005.0932       3.974583  0.8625522996 7.086614 -0.784857299  8.734024
## 2005.0959       3.734267  0.6188388605 6.849696 -1.030369370  8.498904
## 2005.0986       4.358673  1.2398509632 7.477496 -0.411153939  9.128501
## 2005.1014       3.700552  0.5783395269 6.822765 -1.074460095  8.475565
## 2005.1041       3.025586 -0.1000129987 6.151186 -1.754605393  7.805778
## 2005.1068       3.602042  0.4730595454 6.731024 -1.183323681  8.387408
## 2005.1096       4.244309  1.1119472924 7.376671 -0.546224832  9.034843
## 2005.1123       3.860317  0.7245799454 6.996055 -0.935379149  8.656014
## 2005.1151       4.466015  1.3269051891 7.605124 -0.334838954  9.266868
## 2005.1178       3.873607  0.7311287592 7.016084 -0.932398517  8.679612
## 2005.1205       3.712159  0.5663161436 6.858001 -1.098992356  8.523310
## 2005.1233       3.670666  0.5214622525 6.819870 -1.145625568  8.486958
## 2005.1260       3.747240  0.5946783310 6.899801 -1.074186913  8.568667
## 2005.1288       3.922939  0.7670232877 7.078854 -0.903617488  8.749495
## 2005.1315       4.119609  0.9603425139 7.278875 -0.712071910  8.951289
## 2005.1342       4.632859  1.4702458213 7.795472 -0.203940371  9.469658
## 2005.1370       3.196043  0.0300866546 6.361999 -1.645869433  8.037956
## 2005.1397       3.585999  0.4167029089 6.755296 -1.261021207  8.433020
## 2005.1425       3.970228  0.7975956883 7.142861 -0.881894594  8.822351
## 2005.1452       3.526284  0.3503189258 6.702250 -1.330935669  8.383505
## 2005.1479       4.046393  0.8670976760 7.225687 -0.815919381  8.908705
## 2005.1507       5.512837  2.3302166734 8.695458  0.645438999 10.380236
## 2005.1534       5.017057  1.8311137747 8.203000  0.144577319  9.889537
## 2005.1562       4.638323  1.4490603990 7.827585 -0.239233005  9.515878
## 2005.1589       4.060573  0.8679951327 7.253150 -0.822053393  8.943199
## 2005.1616       3.067857 -0.1280325950 6.263747 -1.819834421  7.955549
## 2005.1644       4.305421  1.1062227004 7.504619 -0.587330612  9.198173
## 2005.1671       4.317462  1.1149586764 7.519966 -0.580344312  9.215269
## 2005.1699       3.229872  0.0240667018 6.435677 -1.672984159  8.132728
## 2005.1726       3.596775  0.3876708349 6.805878 -1.311126100  8.504675
## 2005.1753       3.603754  0.3913554469 6.816153 -1.309185769  8.516694
## 2005.1781       5.203353  1.9876626101 8.419044  0.285378901 10.121327
## 2005.1808       5.742314  2.5233347471 8.961292  0.819310326 10.665317
## 2005.1836       4.077321  0.8550569997 7.299584 -0.850706357  9.005348
## 2005.1863       4.062589  0.8370439742 7.288135 -0.870456547  8.995635
## 2005.1890       4.054137  0.8253135547 7.282961 -0.883922366  8.992197
## 2005.1918       4.211363  0.9792648691 7.443462 -0.731704691  9.154431
## 2005.1945       4.080703  0.8453332551 7.316073 -0.867368189  9.028775
## 2005.1973       3.322772  0.0841333699 6.561410 -1.630298209  8.275842
## 2005.2000       3.588025  0.3461219998 6.829929 -1.370037969  8.546089
## 2005.2027       3.397057  0.1518917476 6.642222 -1.565994873  8.360109
## 2005.2055       3.254734  0.0063104616 6.503158 -1.713301076  8.222769
## 2005.2082       3.999101  0.7474219970 7.250779 -0.973912730  8.972114
## 2005.2110       3.685882  0.4309518036 6.940813 -1.292104389  8.663869
## 2005.2137       3.094809 -0.1633700719 6.352989 -1.888146013  8.077764
## 2005.2164       2.764765 -0.4966594277 6.026190 -2.223153403  7.752684
## 2005.2192       4.044633  0.7799659848 7.309300 -0.948244317  9.037510
## 2005.2219       3.374334  0.1064281744 6.642240 -1.623496752  8.372165
## 2005.2247       3.551319  0.2801774781 6.822461 -1.451460374  8.554099
## 2005.2274       4.407270  1.1328956110 7.681644 -0.600453475  9.414993
## 2005.2301       4.319268  1.0416638232 7.596871 -0.693394808  9.331930
## 2005.2329       3.893169  0.6123388729 7.173999 -1.124427621  8.910765
## 2005.2356       3.702789  0.4187357030 6.986842 -1.319736976  8.725315
## 2005.2384       4.461754  1.1744806819 7.749027 -0.565696510  9.489204
## 2005.2411       4.059629  0.7691396835 7.350119 -0.972740352  9.091999
## 2005.2438       3.713391  0.4196880873 7.007095 -1.323893130  8.750676
## 2005.2466       2.779052 -0.5178621889 6.075965 -2.263142929  7.821246
## 2005.2493       3.345216  0.0450950419 6.645337 -1.701883568  8.392316
## 2005.2521       3.849864  0.5465384938 7.153189 -1.202136337  8.901864
## 2005.2548       3.477930  0.1714031764 6.784456 -1.578966232  8.534826
## 2005.2575       3.254087 -0.0556374250 6.563812 -1.807699772  8.315874
## 2005.2603       3.691276  0.3783563263 7.004195 -1.375397325  8.757949
## 2005.2630       4.037317  0.7212058795 7.353429 -1.034237446  9.108872
## 2005.2658       4.357618  1.0383181967 7.676919 -0.718813179  9.434050
## 2005.2685       3.964785  0.6422987525 7.287271 -1.116519053  9.046088
## 2005.2712       3.900952  0.5752836571 7.226621 -1.185218963  8.987124
## 2005.2740       3.783326  0.4544775307 7.112174 -1.307708292  8.874360
## 2005.2767       3.364896  0.0328707813 6.696921 -1.730996639  8.460788
## 2005.2795       3.560533  0.2253347194 6.895732 -1.540212697  8.661279
## 2005.2822       3.535205  0.1968360814 6.873574 -1.570389733  8.640800
## 2005.2849       3.996866  0.6553295358 7.338403 -1.113573085  9.107305
## 2005.2877       3.584990  0.2402888104 6.929691 -1.530289028  8.700269
## 2005.2904       4.173718  0.8258553252 7.521581 -0.946396148  9.293832
## 2005.2932       4.432122  1.0811008366 7.783143 -0.692822692  9.557067
## 2005.2959       3.863826  0.5096494763 7.218003 -1.265944533  8.993597
## 2005.2986       3.575377  0.2180471974 6.932706 -1.559215723  8.709969
## 2005.3014       4.441213  1.0807333740 7.801692 -0.698196891  9.580622
## 2005.3041       4.285115  0.9214885239 7.648741 -0.859107525  9.429337
## 2005.3068       4.075140  0.7083706623 7.441910 -1.073889614  9.224171
## 2005.3096       3.870534  0.5006230612 7.240444 -1.283299889  9.024367
## 2005.3123       3.943879  0.5708305150 7.316928 -1.214753562  9.102512
## 2005.3151       3.751594  0.3754103653 7.127778 -1.411833293  8.915021
## 2005.3178       2.834084 -0.5452321999 6.213399 -2.334133901  8.002301
## 2005.3205       3.534298  0.1518531184 6.916743 -1.638705090  8.707301
## 2005.3233       3.867054  0.4814830109 7.252626 -1.310730174  9.044839
## 2005.3260       3.804647  0.4159523205 7.193342 -1.377914314  8.987208
## 2005.3288       2.973231 -0.4185841751 6.365046 -2.214102736  8.160565
## 2005.3315       2.796357 -0.5985764339 6.191290 -2.395745403  7.988459
## 2005.3342       3.458482  0.0604338113 6.856529 -1.738384052  8.655347
## 2005.3370       3.539810  0.1386498749 6.940970 -1.661815372  8.741435
## 2005.3397       4.363802  0.9595334996 7.768071 -0.842577626  9.570183
## 2005.3425       3.758230  0.3508545325 7.165605 -1.452900969  8.969361
## 2005.3452       3.127244 -0.2832348480 6.537723 -2.088633228  8.343121
## 2005.3479       2.907374 -0.5062052761 6.320953 -2.313245041  8.127993
## 2005.3507       3.905534  0.4888568206 7.322211 -1.319822840  9.130891
## 2005.3534       3.628857  0.2090851180 7.048630 -1.601232952  8.858948
## 2005.3562       3.599975  0.1771107999 7.022840 -1.634844199  8.834795
## 2005.3589       3.759797  0.3338432963 7.185751 -1.479747153  8.999342
## 2005.3616       3.254322 -0.1747185499 6.683363 -1.989942976  8.498587
## 2005.3644       3.533048  0.1009238453 6.965173 -1.715933089  8.782030
## 2005.3671       3.389051 -0.0461542045 6.824257 -1.864642181  8.642745
## 2005.3699       3.453453  0.0151687658 6.891737 -1.804948792  8.711854
## 2005.3726       2.461124 -0.9802357660 5.902483 -2.801981446  7.724229
## 2005.3753       3.253690 -0.1907424021 6.698122 -2.014114752  8.521495
## 2005.3781       3.161180 -0.2863228482 6.608682 -2.111320417  8.433680
## 2005.3808       2.947613 -0.5029568312 6.398183 -2.329578174  8.224804
## 2005.3836       3.285723 -0.1679113593 6.739358 -1.996155033  8.567601
## 2005.3863       3.622941  0.1662443339 7.079637 -1.663620232  8.909502
## 2005.3890       3.678382  0.2186259766 7.138137 -1.612858048  8.969621
## 2005.3918       3.070614 -0.3921986738 6.533426 -2.225300726  8.366528
## 2005.3945       3.293574 -0.1722917453 6.759440 -2.007010398  8.594159
## 2005.3973       3.300423 -0.1684941918 6.769340 -2.004828022  8.605674
## 2005.4000       3.755199  0.2832331848 7.227164 -1.554714404  9.065112
## 2005.4027       3.529622  0.0546105132 7.004633 -1.784949418  8.844193
## 2005.4055       2.987846 -0.4902085762 6.465901 -2.331379438  8.307071
## 2005.4082       2.847565 -0.6335304910 6.328660 -2.476310875  8.171440
## 2005.4110       2.973952 -0.5101812256 6.458084 -2.354569727  8.302473
## 2005.4137       3.396945 -0.0902229122 6.884113 -1.936218130  8.730108
## 2005.4164       3.526150  0.0359497291 7.016351 -1.811650808  8.863951
## 2005.4192       3.086001 -0.4072295355 6.579231 -2.256433999  8.428436
## 2005.4219       3.110623 -0.3856345922 6.606881 -2.236441592  8.457688
## 2005.4247       2.457786 -1.0414965905 5.957068 -2.893904739  7.809476
## 2005.4274       2.625129 -0.8771755268 6.127433 -2.731183443  7.981441
## 2005.4301       2.943813 -0.5615109796 6.449137 -2.417117283  8.304743
## 2005.4329       2.469123 -1.0392176178 5.977464 -2.896420933  7.834667
## 2005.4356       2.449614 -1.0617406992 5.960969 -2.920539654  7.819768
## 2005.4384       2.973090 -0.5412764092 6.487457 -2.401669636  8.347850
## 2005.4411       2.703488 -0.8138877237 6.220863 -2.675873856  8.082849
## 2005.4438       2.572985 -0.9473972327 6.093367 -2.810974910  7.956945
## 2005.4466       2.819954 -0.7034314773 6.343340 -2.568599341  8.208508
## 2005.4493       2.973583 -0.5528041278 6.499971 -2.419560824  8.366727
## 2005.4521       3.077371 -0.4520155765 6.606757 -2.320359754  8.475101
## 2005.4548       2.912194 -0.6201884704 6.444576 -2.490118781  8.314507
## 2005.4575       3.009123 -0.5262529150 6.544499 -2.397768014  8.416015
## 2005.4603       3.075898 -0.4624691864 6.614266 -2.335567734  8.487364
## 2005.4630       3.384883 -0.1564733700 6.926239 -2.031154028  8.800919
## 2005.4658       3.143759 -0.4005829031 6.688102 -2.276844338  8.564363
## 2005.4685       2.772826 -0.7745001063 6.320152 -2.652340987  8.197993
## 2005.4712       3.111874 -0.4384325966 6.662181 -2.317851596  8.541600
## 2005.4740       3.144519 -0.4087664938 6.697805 -2.289762287  8.578801
## 2005.4767       3.527727 -0.0285352675 7.083988 -1.911106535  8.966560
## 2005.4795       3.293097 -0.2661383132 6.852333 -2.150283737  8.736478
## 2005.4822       2.743152 -0.8190549475 6.305358 -2.704773214  8.191077
## 2005.4849       2.983511 -0.5816641246 6.548686 -2.468953922  8.435976
## 2005.4877       3.031971 -0.5361703074 6.600113 -2.425030329  8.488973
## 2005.4904       3.146489 -0.4246166322 6.717594 -2.315045574  8.608023
## 2005.4932       2.952027 -0.6220399623 6.526093 -2.514036523  8.418090
## 2005.4959       3.394896 -0.1821296691 6.971921 -2.075692551  8.865484
## 2005.4986       3.195302 -0.3846800225 6.775284 -2.279807930  8.670412
## 2005.5014       3.141474 -0.4414620443 6.724410 -2.338153687  8.621101
## 2005.5041       2.923191 -0.6626960589 6.509079 -2.560950149  8.407333
## 2005.5068       3.155422 -0.4334143015 6.744259 -2.333229553  8.644074
## 2005.5096       3.016617 -0.5751663486 6.608400 -2.476541481  8.509775
## 2005.5123       2.792930 -0.8017971895 6.387658 -2.704730923  8.290591
## 2005.5151       3.156801 -0.4408685170 6.754470 -2.345359576  8.658961
## 2005.5178       3.072207 -0.5284013452 6.672816 -2.434448458  8.578863
## 2005.5205       3.031100 -0.5724460812 6.634645 -2.480047978  8.542247
## 2005.5233       3.339231 -0.2672493270 6.945712 -2.176404742  8.854867
## 2005.5260       3.010581 -0.5988321307 6.619993 -2.509539801  8.530701
## 2005.5288       3.358299 -0.2540435060 6.970642 -2.166302171  8.882900
## 2005.5315       2.798293 -0.8169766632 6.413564 -2.730785066  8.327372
## 2005.5342       3.084867 -0.5333281148 6.703062 -2.448685002  8.618419
## 2005.5370       3.279240 -0.3418780749 6.900358 -2.258782195  8.817262
## 2005.5397       3.505831 -0.1182071222 7.129870 -2.036657228  9.048320
## 2005.5425       3.148788 -0.4781685894 6.775745 -2.398163435  8.695739
## 2005.5452       3.648061  0.0181885878 7.277933 -1.903349757  9.199472
## 2005.5479       3.047522 -0.5852640756 6.680307 -2.508344680  8.603388
## 2005.5507       3.415722 -0.2199745065 7.051419 -2.144596135  8.976041
## 2005.5534       3.036535 -0.6020701124 6.675141 -2.528231532  8.601302
## 2005.5562       2.952802 -0.6887100133 6.594314 -2.616409994  8.522014
## 2005.5589       3.228395 -0.4160208034 6.872811 -2.345258118  8.802049
## 2005.5616       3.511464 -0.1358533821 7.158782 -2.066626807  9.089556
## 2005.5644       3.180136 -0.4700816723 6.830353 -2.402389986  8.762661
## 2005.5671       2.898319 -0.7547950135 6.551434 -2.688636998  8.485276
## 2005.5699       3.356230 -0.2997790638 7.012240 -2.235153504  8.947614
## 2005.5726       3.154182 -0.5047203176 6.813084 -2.441626000  8.749989
## 2005.5753       3.758840  0.0970478529 7.420632 -1.841387863  9.359068
## 2005.5781       3.228186 -0.4364938522 6.892867 -2.376458395  8.832831
## 2005.5808       3.349316 -0.3182504867 7.016882 -2.259742652  8.958374
## 2005.5836       3.264179 -0.4062708220 6.934628 -2.349289409  8.877647
## 2005.5863       3.044852 -0.6284790776 6.718182 -2.573022888  8.662726
## 2005.5890       3.034840 -0.6413699188 6.711049 -2.587437757  8.657117
## 2005.5918       3.281519 -0.3975673121 6.960605 -2.345157986  8.908196
## 2005.5945       2.987035 -0.6949262048 6.668995 -2.644038525  8.618108
## 2005.5973       3.097615 -0.5872175940 6.782448 -2.537850372  8.733081
## 2005.6000       3.784652  0.0969486829 7.472355 -1.855203370  9.424507
## 2005.6027       4.038829  0.3482578444 7.729399 -1.605412302  9.683070
## 2005.6055       3.444853 -0.2485835089 7.138289 -2.203770569  9.093476
## 2005.6082       2.776498 -0.9198013983 6.472798 -2.876504197  8.429501
## 2005.6110       2.948794 -0.7503669090 6.647954 -2.708584273  8.606172
## 2005.6137       3.579169 -0.1228507133 7.281188 -2.082581471  9.240919
## 2005.6164       3.220736 -0.4841406163 6.925612 -2.445383601  8.886855
## 2005.6192       3.293723 -0.4140073081 7.001454 -2.376761355  8.964208
## 2005.6219       3.004188 -0.7063945951 6.714771 -2.670658541  8.679035
## 2005.6247       2.895269 -0.8181641856 6.608702 -2.783936871  8.574474
## 2005.6274       2.825361 -0.8909197103 6.541642 -2.858199978  8.508922
## 2005.6301       2.892591 -0.8265355943 6.611718 -2.795322290  8.580504
## 2005.6329       3.508446 -0.2135237846 7.230416 -2.183815756  9.200708
## 2005.6356       3.325353 -0.3994589211 7.050164 -2.371255020  9.021960
## 2005.6384       3.523310 -0.2043408492 7.250961 -2.177639929  9.224260
## 2005.6411       2.868934 -0.8615534147 6.599422 -2.836354331  8.574223
## 2005.6438       3.259458 -0.4738645980 6.992781 -2.450166209  8.969082
## 2005.6466       3.031638 -0.7045176115 6.767793 -2.682318780  8.745594
## 2005.6493       3.793565  0.0545788455 7.532551 -1.924720743  9.511850
## 2005.6521       3.725934 -0.0158801540 7.467749 -1.996677030  9.448545
## 2005.6548       3.205093 -0.5395472246 6.949734 -2.521840257  8.932027
## 2005.6575       2.863936 -0.8835288040 6.611401 -2.867316864  8.595189
## 2005.6603       3.304459 -0.4458278725 7.054746 -2.431109834  9.040028
## 2005.6630       3.155843 -0.5972642063 6.908949 -2.584038947  8.895724
## 2005.6658       2.840049 -0.9158754889 6.595974 -2.904141887  8.584240
## 2005.6685       2.707848 -1.0508923080 6.466588 -3.040649246  8.456345
## 2005.6712       2.795132 -0.9664223511 6.556685 -2.957668713  8.547932
## 2005.6740       3.059165 -0.7052000424 6.823531 -2.697934715  8.816265
## 2005.6767       3.324944 -0.4422304421 7.092119 -2.436452314  9.086341
## 2005.6795       3.290368 -0.4796144793 7.060350 -2.475322443  9.056058
## 2005.6822       3.332360 -0.4404276544 7.105147 -2.437620604  9.102340
## 2005.6849       3.567930 -0.2076603564 7.343520 -2.206337189  9.342197
## 2005.6877       3.691913 -0.0864783402 7.470305 -2.086637954  9.470464
## 2005.6904       3.241411 -0.5397798735 7.022601 -2.541421171  9.024242
## 2005.6932       3.113256 -0.6707312587 6.897243 -2.673853143  8.900365
## 2005.6959       3.496012 -0.2907706032 7.282794 -2.295371981  9.287395
## 2005.6986       3.445482 -0.3440929606 7.235057 -2.350172741  9.241137
## 2005.7014       3.180317 -0.6120481359 6.972683 -2.619605230  8.980240
## 2005.7041       2.915929 -0.8792256344 6.711083 -2.888258956  8.720116
## 2005.7068       2.697170 -1.1007709141 6.495111 -3.111279380  8.505619
## 2005.7096       3.285659 -0.5150668275 7.086384 -2.527049355  9.098367
## 2005.7123       2.625101 -1.1784070097 6.428609 -3.191862520  8.442064
## 2005.7151       2.654321 -1.1519675740 6.460609 -3.166894991  8.475537
## 2005.7178       3.179420 -0.6296470924 6.988487 -2.646045341  9.004885
## 2005.7205       3.334191 -0.4776528698 7.146034 -2.495520878  9.163902
## 2005.7233       3.125878 -0.6887395110 6.940496 -2.708076209  8.959833
## 2005.7260       3.821928  0.0045379240 7.639318 -2.016266396  9.660123
## 2005.7288       3.536404 -0.2837569501 7.356564 -2.306027827  9.378835
## 2005.7315       3.239520 -0.5834087285 7.062449 -2.607145100  9.086186
## 2005.7342       3.905553  0.0798572812 7.731248 -1.945343525  9.756449
## 2005.7370       3.479877 -0.3485829028 7.308337 -2.375247085  9.335001
## 2005.7397       3.201934 -0.6292884650 7.033156 -2.657414967  9.061282
## 2005.7425       3.228888 -0.6050941968 7.062871 -2.634681965  9.092459
## 2005.7452       3.152221 -0.6845197313 6.988962 -2.715567714  9.020010
## 2005.7479       3.900346  0.0608489698 7.739844 -1.971658179  9.772351
## 2005.7507       2.944691 -0.8975603632 6.786943 -2.931525632  8.820909
## 2005.7534       3.673922 -0.1710825232 7.518926 -2.206504866  9.554348
## 2005.7562       3.847909  0.0001536754 7.695663 -2.036724700  9.732542
## 2005.7589       3.778530 -0.0719734894 7.629033 -2.110306857  9.667367
## 2005.7616       3.115121 -0.7381292555 6.968371 -2.777916577  9.008158
## 2005.7644       3.357914 -0.4980810282 7.213908 -2.539321269  9.255148
## 2005.7671       3.465230 -0.3935076841 7.323967 -2.436199810  9.366659
## 2005.7699       3.754470 -0.1070075652 7.615948 -2.151150545  9.660091
## 2005.7726       3.840091 -0.0241254872 7.704308 -2.069718292  9.749901
## 2005.7753       3.983698  0.1167440623 7.850651 -1.930297541  9.897693
## 2005.7781       3.932843  0.0631547611 7.802532 -1.985334616  9.851021
## 2005.7808       3.782285 -0.0901369659 7.654706 -2.140073094  9.704642
## 2005.7836       3.325113 -0.5500395226 7.200266 -2.601421381  9.251647
## 2005.7863       3.301687 -0.5761947073 7.179569 -2.629021278  9.232395
## 2005.7890       3.602056 -0.2785529695 7.482665 -2.332823237  9.536935
## 2005.7918       3.280751 -0.6025833210 7.164085 -2.658296271  9.219798
## 2005.7945       3.376233 -0.5098246029 7.262291 -2.566979224  9.319445
## 2005.7973       3.557491 -0.3312881344 7.446270 -2.389883417  9.504865
## 2005.8000       4.054772  0.1632738099 7.946271 -1.896761126 10.006306
## 2005.8027       3.839486 -0.0547304856 7.733702 -2.116204070  9.795176
## 2005.8055       3.100940 -0.7959924505 6.997872 -2.858903680  9.060783
## 2005.8082       3.573701 -0.3259444787 7.473347 -2.390292352  9.537695
## 2005.8110       3.822717 -0.0796405466 7.725075 -2.145424064  9.790859
## 2005.8137       3.463569 -0.4414995475 7.368637 -2.508717713  9.435855
## 2005.8164       3.313529 -0.5942469903 7.221306 -2.662898809  9.289957
## 2005.8192       3.821311 -0.0891712412 7.731794 -2.159255719  9.801879
## 2005.8219       3.742743 -0.1704443990 7.655930 -2.241960546  9.727446
## 2005.8247       2.114504 -1.8013859517 6.030394 -3.874332779  8.103340
## 2005.8274       2.346998 -1.5715922381 6.265589 -3.645968759  8.339965
## 2005.8301       3.099915 -0.8213746528 7.021204 -2.897179883  9.097009
## 2005.8329       3.050182 -0.8738047746 6.974168 -2.951037731  9.051401
## 2005.8356       2.909701 -1.0169810178 6.836382 -3.095640719  8.915042
## 2005.8384       3.267784 -0.6615908605 7.197159 -2.741676329  9.277245
## 2005.8411       3.446398 -0.4856681162 7.378465 -2.567178375  9.459975
## 2005.8438       3.057415 -0.8773414066 6.992171 -2.960275481  9.075105
## 2005.8466       3.523398 -0.4140455495 7.460842 -2.498402467  9.545199
## 2005.8493       3.797508 -0.1426216568 7.737638 -2.228400447  9.823417
## 2005.8521       2.847015 -1.0957987704 6.789829 -3.182998465  8.877029
## 2005.8548       2.991343 -0.9541531348 6.936840 -3.042772766  9.025459
## 2005.8575       3.716269 -0.2319081126 7.664446 -2.321946717  9.754484
## 2005.8603       3.934839 -0.0160161813 7.885695 -2.107472795  9.977152
## 2005.8630       4.406596  0.4530634181 8.360128 -1.639810245 10.453002
## 2005.8658       3.526536 -0.4296713781 7.482744 -2.523961131  9.577033
## 2005.8685       3.837702 -0.1211783924 7.796583 -2.216883279  9.892288
## 2005.8712       3.596493 -0.3650591735 7.558045 -2.462178238  9.655164
## 2005.8740       3.857116 -0.1071061238 7.821338 -2.205638414  9.919870
## 2005.8767       3.906089 -0.0608009698 7.872978 -2.160745535  9.972923
## 2005.8795       3.270415 -0.6991406786 7.239971 -2.800496569  9.341327
## 2005.8822       3.333994 -0.6382261969 7.306214 -2.740992465  9.408980
## 2005.8849       4.122678  0.1477950401 8.097560 -1.956380661 10.201736
## 2005.8877       3.804409 -0.1731340201 7.781952 -2.278718210  9.887537
## 2005.8904       3.789673 -0.1905296361 7.769875 -2.297521374  9.876866
## 2005.8932       3.714606 -0.2682537611 7.697465 -2.376652107  9.805863
## 2005.8959       3.560560 -0.4249547340 7.546075 -2.534758750  9.655879
## 2005.8986       3.547938 -0.4402306503 7.536106 -2.551439400  9.647315
## 2005.9014       3.972143 -0.0186773146 7.962963 -2.131289865 10.075575
## 2005.9041       3.755462 -0.2380086099 7.748932 -2.352024028  9.862947
## 2005.9068       3.448248 -0.5478708836 7.444366 -2.663288240  9.559783
## 2005.9096       4.557044  0.5582788895 8.555809 -1.558539476 10.672627
## 2005.9123       5.271880  1.2704705741 9.273290 -0.847747874 11.391509
## 2005.9151       5.449443  1.4453896494 9.453496 -0.674227956 11.573113
## 2005.9178       3.386854 -0.6198406202 7.393548 -2.740856461  9.514564
## 2005.9205       4.479393  0.4700595500 8.488727 -1.652353604 10.611140
## 2005.9233       4.873154  0.8611823952 8.885126 -1.262627153 11.008935
## 2005.9260       4.640047  0.6254390932 8.654655 -1.499765932 10.779860
## 2005.9288       4.375590  0.3583474845 8.392832 -1.768252102 10.519432
## 2005.9315       5.339619  1.3197439159 9.359494 -0.808249318 11.487487
## 2005.9342       3.647156 -0.3753495426 7.669662 -2.504735511  9.799048
## 2005.9370       3.993534 -0.0316015186 8.018669 -2.162379312 10.149446
## 2005.9397       3.401641 -0.6261215839 7.429404 -2.758290294  9.561572
## 2005.9425       3.873255 -0.1571330523 7.903644 -2.290691771 10.037202
## 2005.9452       3.798746 -0.2342662713 7.831759 -2.369214095  9.966706
## 2005.9479       4.834415  0.7987800559 8.870050 -1.337555969 11.006386
## 2005.9507       4.485037  0.4467810860 8.523292 -1.690942239 10.661015
## 2005.9534       4.628231  0.5873570361 8.669106 -1.551752688 10.808216
## 2005.9562       3.974883 -0.0686086504 8.018375 -2.209103877 10.158870
## 2005.9589       3.930952 -0.1151552626 7.977059 -2.257035095 10.118939
## 2005.9616       4.343130  0.2944088533 8.391851 -1.848854690 10.535115
## 2005.9644       3.899447 -0.1518859711 7.950781 -2.296532333 10.095427
## 2005.9671       4.009766 -0.0441781026 8.063710 -2.190206392 10.209738
## 2005.9699       4.319703  0.2631504770 8.376256 -1.884258851 10.523665
## 2005.9726       3.333536 -0.7256236511 7.392696 -2.874413130  9.541486
## 2005.9753       4.017467 -0.0442988576 8.079232 -2.194467601 10.229401
## 2005.9781       4.172191  0.1078220493 8.236561 -2.043725074 10.388108
## 2005.9808       4.346757  0.2797858856 8.413729 -1.873138736 10.566653
## 2005.9836       4.706885  0.6373135743 8.776457 -1.516987665 10.930759
## 2005.9863       4.760269  0.6880985886 8.832440 -1.467578389 10.988117
## 2005.9890       4.496551  0.4217830165 8.571319 -1.735268821 10.728371
## 2005.9918       4.115070  0.0377064567 8.192433 -2.120719366 10.350859
## 2005.9945       3.541175 -0.5387826662 7.621132 -2.698581600  9.780931
## 2005.9973       3.419677 -0.6628724501 7.502227 -2.824043622  9.663398
## 2006.0000       3.636529 -0.4486110401 7.721669 -2.611153580  9.884212
## 2006.0027       4.057684 -0.0300447444 8.145413 -2.193957783 10.309326
## 2006.0055       4.086440 -0.0038765619 8.176756 -2.169159232 10.342039

NOTE: #AIC = 16371.83 , BIC = 16388.91, RMSE = 0.8912478 Also, the headline of the plot below contains ETS(A,N,N) showing that the automated model characterized error of the “tsWSR_PK” SERIES as Additative (A), trend as None (A) and seasonality as None (N) which is equivalent to simple exponential model(ses) with only alpha parameter i.e level.

#Model 2 : ARIMA Models(AutoRegressive Moving Average)

It is used for forecasting for both seasonal and non- seasonal level.

#Changing to stationary set

train.sta <- diff(train)
test.sta <- diff(test)

2.1 Auto Arima model without sesonality:

set.seed(301)
model <- auto.arima(train.sta, seasonal = FALSE)
summary(model)
## Series: train.sta 
## ARIMA(1,0,3) with zero mean 
## 
## Coefficients:
##           ar1     ma1      ma2      ma3
##       -0.8918  0.2659  -0.8286  -0.2770
## s.e.   0.0745  0.0753   0.0529   0.0247
## 
## sigma^2 estimated as 0.9689:  log likelihood=-3074.56
## AIC=6159.11   AICc=6159.14   BIC=6187.58
## 
## Training set error measures:
##                        ME      RMSE       MAE MPE MAPE      MASE        ACF1
## Training set -0.006952857 0.9834439 0.7398732 NaN  Inf 0.5795203 0.005148871

NOTE: #AIC = 6159.11 , BIC = 6187.58, RMSE = 0.9834439

tsdisplay(residuals(model), lag.max = 45, main = '(1,1,1) Model Residuals')

2.2. Auto Arima model with sesonality:

set.seed(301)
model2 <- auto.arima(train.sta, seasonal= TRUE)
summary(model2)
## Series: train.sta 
## ARIMA(1,0,3) with zero mean 
## 
## Coefficients:
##           ar1     ma1      ma2      ma3
##       -0.8918  0.2659  -0.8286  -0.2770
## s.e.   0.0745  0.0753   0.0529   0.0247
## 
## sigma^2 estimated as 0.9689:  log likelihood=-3074.56
## AIC=6159.11   AICc=6159.14   BIC=6187.58
## 
## Training set error measures:
##                        ME      RMSE       MAE MPE MAPE      MASE        ACF1
## Training set -0.006952857 0.9834439 0.7398732 NaN  Inf 0.5795203 0.005148871

NOTE: #AIC = 6159.11 , BIC = 6187.58, RMSE = 0.9834439

tsdisplay(residuals(model2), lag.max = 20, main = 'Seasonal Model Residuals')

NOTE: Above Graphs shows serious lags at 17, so modify model for p or q = 17

  1. Manual ARIMA:
    Passing parameter order = (1,0,18) which means p(member of lags of a variable to be used as a predictor) = 1, d = 0(members of differences needed for stationarity) and q = 18 (moving average order - number of lagged forecasts error in the prediction equation)
set.seed(301)
model1 <- arima(train.sta, order = c(1,0,18))
summary(model1)
## 
## Call:
## arima(x = train.sta, order = c(1, 0, 18))
## 
## Coefficients:
##          ar1      ma1      ma2      ma3     ma4      ma5     ma6      ma7
##       0.0870  -0.7079  -0.1979  -0.0288  0.0159  -0.0087  0.0193  -0.0193
## s.e.  0.3104   0.3097   0.1940   0.0810  0.0309   0.0267  0.0265   0.0273
##          ma8      ma9    ma10     ma11     ma12     ma13    ma14    ma15
##       0.0246  -0.0385  0.0548  -0.0159  -0.0182  -0.0287  0.0227  0.0300
## s.e.  0.0272   0.0269  0.0284   0.0307   0.0272   0.0267  0.0298  0.0272
##          ma16     ma17    ma18  intercept
##       -0.0258  -0.0647  0.0573    -0.0005
## s.e.   0.0285   0.0281  0.0235     0.0016
## 
## sigma^2 estimated as 0.9595:  log likelihood = -3065.9,  aic = 6173.8
## 
## Training set error measures:
##                        ME      RMSE       MAE MPE MAPE      MASE          ACF1
## Training set -0.001369022 0.9795425 0.7387716 NaN  Inf 0.5469141 -9.531875e-05

NOTE: #AIC = 6173.8 , RMSE = 0.9795425

tsdisplay(residuals(model1), lag.max = 20, main = 'Seasonal Model Residuals')

Note: It seems good fit.

Fit the observed with the predicted values for both method models in the same plot.

par(mfrow = c(2,2))
fit_auto_train %>% forecast(h=341) %>% autoplot() + autolayer(test)

model1 %>% forecast(h=341) %>% autoplot() 

model2 %>% forecast(h=341) %>% autoplot() 

model %>% forecast(h=341) %>% autoplot()  

COMMENT:
1. The plots includes 80% to 95% confidence interval.
2. For ARIMA models it plots the forecast but because of the nature of autoregressive process of 1 lag i.e. AR1, these forecast stabilizes quite smoothly.

Plot and comment on the residuals of the fitted data for both models in the same plot.

Residual Plot - to confirm no problem with this model

par(mfrow= c(2,2))
hist(fit_auto_train$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model$residuals))

hist(model$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model$residuals))

hist(model1$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model$residuals))

hist(model2$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model$residuals))

OBSERVATION: The residuals for each model are normally distributed, which shows the good fit of the models. It is normally distrbuted.

Testing the autocorrelation and partial autocorrelation of the residuals (using the plots and Ljung-Box Q statistic) up to an appropriate lag *

  1. ACF & PACF Plots

ACF(Autocorelation functions): These gives number of moving average(MA) terms in the model.
PACF(Partial Autocorelation Factors): There gives the number of AR terms in the model.

par(mfrow = c(2,2))
acf(fit_auto_train$residuals, main = 'Correlogram')
acf(model$residuals, main = 'Correlogram')
acf(model1$residuals, main = 'Correlogram')
acf(model2$residuals, main = 'Correlogram')

par(mfrow = c(2,2))
pacf(fit_auto_train$residuals, main = 'Partial Corelogram')
pacf(model$residuals, main = 'Partial Corelogram')
pacf(model1$residuals, main = 'Partial Corelogram')
pacf(model2$residuals, main = 'Partial Corelogram')

NOTE:
1. Residuals are the difference between actaual and fitted values
2. These blue lines are signficance bounds.
3. This graph shows us the autocorelations for insample forecast errors
4. Do not exceed these significance bounds for lags 1 to end

Here, our model produce residuals that are normal.It is not exhibiting any pattern or structure, this sums to have a consistant variance.

  1. Ljunx-Box Test to check the autocorelations of the residuals

This test used to test the lack of fit of a time series model. This test is applied to the residuals of the time series model

HO: No serial correlation upto 20 lags (does not exhibit lack of fit of the model) Ha: Serial corelation is present (the model exhibits lack of fit)

We reject the null hypothesis, if p value is less than 0.05. This test is generally done for ARIMA models

Box.test(model$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model$residuals
## X-squared = 16.716, df = 20, p-value = 0.6713
Box.test(model1$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model1$residuals
## X-squared = 1.1949, df = 20, p-value = 1
Box.test(model2$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model2$residuals
## X-squared = 16.716, df = 20, p-value = 0.6713

OBSERVATIONS:
For above performed Ljunx-Box Test on model,model1 and model2 output, we can say that p-value is more than significance level 0.05, then there is not a statistical significance. We conclude that there is a liitle evidence of non-zero auto-corelations in the insample forecast errors at lags 1 to 20.
So we fail to rejects the null hypothesis. Hence, we conclude that there is no serial auto corelation present between lags. Hence, the model seems to be good fit.

Test and comment on the normality of the residuals.

Here, further testing the normality by plotting Normal Q-Q plots. From below plots we observed that most of the datapoints lies on the line. Hence, we could conclude that our model produces residuals that are normal

par(mfrow = c(2,2))
qqnorm(fit_auto_train$residuals)
qqnorm(model$residuals)
qqnorm(model1$residuals)
qqnorm(model2$residuals)

Validating The Model by testing the model performance with Holdout set

Here, choosing RMSE as evaluation parameter. The lower RMSE indicates a more accurate forecast.

Below, is the comparison of model. We are fitting the model to test set which is the holdout set. And checking the accuracy using RMSE(Root mean square error) evaluation parameter.

accuracy(forecast(fit_auto_train), test) ["Test set", "RMSE"]
## [1] 1.273901
#o/p - 1.273901
accuracy(forecast(model1), test.sta) ["Test set", "RMSE"]
## [1] 1.253145
#O/P - 1.253145
accuracy(forecast(model2), test.sta) ["Test set", "RMSE"]
## [1] 1.253728
#o/p - 1.253728
accuracy(forecast(model), test.sta) ["Test set", "RMSE"]
## [1] 1.253728
#O/P - 1.253728

NOTE:- Here, the RMSE value is less of ARIMA models compared to exponential model.
The minimum RMSE is of model1.
The RMSE of auto arima models with seasonality and without seasonality have no difference. It gives the same result for this series.

COMMENT ON BEST MODEL AND ITS PARAMETERS We observed the best model to be ARIMA model i.e.“model1”.

summary(model1)
## 
## Call:
## arima(x = train.sta, order = c(1, 0, 18))
## 
## Coefficients:
##          ar1      ma1      ma2      ma3     ma4      ma5     ma6      ma7
##       0.0870  -0.7079  -0.1979  -0.0288  0.0159  -0.0087  0.0193  -0.0193
## s.e.  0.3104   0.3097   0.1940   0.0810  0.0309   0.0267  0.0265   0.0273
##          ma8      ma9    ma10     ma11     ma12     ma13    ma14    ma15
##       0.0246  -0.0385  0.0548  -0.0159  -0.0182  -0.0287  0.0227  0.0300
## s.e.  0.0272   0.0269  0.0284   0.0307   0.0272   0.0267  0.0298  0.0272
##          ma16     ma17    ma18  intercept
##       -0.0258  -0.0647  0.0573    -0.0005
## s.e.   0.0285   0.0281  0.0235     0.0016
## 
## sigma^2 estimated as 0.9595:  log likelihood = -3065.9,  aic = 6173.8
## 
## Training set error measures:
##                        ME      RMSE       MAE MPE MAPE      MASE          ACF1
## Training set -0.001369022 0.9795425 0.7387716 NaN  Inf 0.5469141 -9.531875e-05
model1$coef
##          ar1          ma1          ma2          ma3          ma4          ma5 
##  0.087015549 -0.707885319 -0.197873344 -0.028846474  0.015867352 -0.008672255 
##          ma6          ma7          ma8          ma9         ma10         ma11 
##  0.019276027 -0.019316858  0.024581787 -0.038470988  0.054800729 -0.015859729 
##         ma12         ma13         ma14         ma15         ma16         ma17 
## -0.018230722 -0.028693866  0.022711597  0.030005760 -0.025773729 -0.064668514 
##         ma18    intercept 
##  0.057287130 -0.000485985

Passing parameter order = (1,0,18) ,which means

p = 1, d = 0 q = 18

For the best model, 1 member of lag is used as a predictor. As we transformed the series to stationary so members of differences needed for stationarity is 0. 18 are the number of lagged forecasts errors used in the prediction equation. AIC = 6173.8, RMSE = 0.9795425 which is minimum as compared to other models.

FOR VARIABLE 2

2. Forecasting on Peak T (T_PK)

#Checking the attributes of time series

attributes(tsT_PK)
## $tsp
## [1] 1998.00 2004.94  365.00
## 
## $class
## [1] "ts"

#Plotting the series

plot(tsT_PK, main = "Peak tsT_PK", col = "red")

From the plot, we observes that there is no trend but seems to have seasonlity.
Most of the data is missing for year 2002 to 2003.

Checking the statistics and visualising the missing data in the series:

plotNA.distribution(tsT_PK)

As we see there is a large gap in the missing data, so locf and NOCB would not seems the good technique to implement here.

Hence, using “Linear interpolation” technique to impute missing values. This method works good for series having seasonality but no trend type of time series data.

#Impute the missing values with “na_interpolation” function of “imputeTS package” and visualise the imputed values in the time series

tsT_PK.imp <- na_interpolation(tsT_PK, method = "linear")
plotNA.imputations(tsT_PK, tsT_PK.imp)

#Decomposition Of Time Series

decomp1 <- decompose(tsT_PK.imp)
plot(decomp1, col = "red")

OBSERVATIONS:
1)There does not seems to have any trend.
2)We observe some seasonality in the series. Means there is some seasonality. Seasonality describes cyclical effects due to the time of the year.

Let’s explore this seasonality pattern by plotting the decomp$seasonal components such that y-axis represents Seasonality Index and x-axis.

#Visualising the seasonal component

par(mfrow=c(1,2))
plot(decomp1$seasonal, type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")
plot(decomp1$seasonal[1:400], type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")

OBSERVATION
1) There is increase in the mid of months of a year.
2) There is a drop in peak T for starting and ending months of the year

#Stationarity check of time series

We could check the stationarity of time series by ADF and KPSS Tests:

Augmented Dickey- Fuller(ADF) t-statistic test for unit root and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) for level or trend stationarity Tests:

adf.test(tsT_PK.imp)
## 
##  Augmented Dickey-Fuller Test
## 
## data:  tsT_PK.imp
## Dickey-Fuller = -3.8863, Lag order = 13, p-value = 0.01467
## alternative hypothesis: stationary
##It results stationary, as p-value is less than 0.05
kpss.test(tsT_PK.imp, null="Trend")
## 
##  KPSS Test for Trend Stationarity
## 
## data:  tsT_PK.imp
## KPSS Trend = 0.18987, Truncation lag parameter = 8, p-value = 0.0198

NOTE: It results non- stationary as p-value is less than 0.05. More p-value means stationary series

par(mfrow=c(1,1))
acf(tsT_PK.imp,lag.max = length(tsT_PK.imp),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

#Here, most of the lags are outside the significance bound, which means data is not stationary

#difference of 2 is sufficient

acf(diff(tsT_PK.imp, diff = 2))

Here, we observes most of the data lies outside the blue-dotted lines means the significance level. Now, we will take the difference to make our data stationary.

It seems difference of 2 is sufficient to make the series stationary

tsT_PK.sta <- diff(tsT_PK.imp, diff = 2)
attributes(tsT_PK.sta)
## $tsp
## [1] 1998.005 2004.940  365.000
## 
## $class
## [1] "ts"
acf(tsT_PK.sta, lag.max = length(tsT_PK.sta),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Hence, after differencing our series is now transformed to stationary time series.

#Visualising the stationary series

plot(tsT_PK.sta, col = "red")

CREATING MODELS AND FORECASTING: USING EXPONENTIAL SMOOTHING AND ARIMA TECHNIQUES FOR TIME SERIES FORECASTING

1.Train - Test Split of the data

train1 <- window(tsT_PK.imp, end = c(2004, 3))
test1 <- window(tsT_PK.imp, start = c(2004, 4))

#Model 1 : Auto Exponential Smoothening Model

set.seed(301)
fit_auto_train_VAR2 <- forecast(train1)
summary(fit_auto_train_VAR2)
## 
## Forecast method: STL +  ETS(A,N,N)
## 
## Model Information:
## ETS(A,N,N) 
## 
## Call:
##  ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend) 
## 
##   Smoothing parameters:
##     alpha = 0.8786 
## 
##   Initial states:
##     l = 25.3318 
## 
##   sigma:  2.9034
## 
##      AIC     AICc      BIC 
## 21549.73 21549.74 21566.81 
## 
## Error measures:
##                       ME     RMSE      MAE       MPE     MAPE      MASE
## Training set 0.002767155 2.902054 2.029788 -2.028444 11.01258 0.4517796
##                    ACF1
## Training set 0.02845628
## 
## Forecasts:
##           Point Forecast        Lo 80     Hi 80         Lo 95     Hi 95
## 2004.0082       21.33776  17.61692695  25.05859   15.64723876  27.02827
## 2004.0110       22.07243  17.11954862  27.02532   14.49764993  29.64722
## 2004.0137       20.91589  14.98145198  26.85033   11.83995012  29.99183
## 2004.0164       24.55218  17.77692819  31.32743   14.19032603  34.91404
## 2004.0192       21.65143  14.12876170  29.17410   10.14650151  33.15636
## 2004.0219       19.43034  11.22808227  27.63260    6.88606921  31.97461
## 2004.0247       24.41696  15.58726690  33.24666   10.91310830  37.92082
## 2004.0274       25.88847  16.47305495  35.30388   11.48883588  40.28810
## 2004.0301       26.02004  16.05326952  35.98681   10.77717997  41.26290
## 2004.0329       24.37379  13.88460805  34.86298    8.33196885  40.41562
## 2004.0356       22.24627  11.25948520  33.23306    5.44343128  39.04911
## 2004.0384       23.14929  11.68647891  34.61210    5.61843438  40.68014
## 2004.0411       22.54337  10.62352985  34.46320    4.31355001  40.77318
## 2004.0438       21.98502   9.62504852  34.34500    3.08207315  40.88797
## 2004.0466       22.87562  10.09065104  35.66059    3.32269655  42.42855
## 2004.0493       21.75903   8.56274360  34.95532    1.57705185  41.94101
## 2004.0521       20.15676   6.56160118  33.75193   -0.63524329  40.94877
## 2004.0548       23.04116   9.05849259  37.02382    1.65651640  44.42580
## 2004.0575       24.01153   9.65181564  38.37125    2.05024128  45.97282
## 2004.0603       22.03129   7.30417561  36.75841   -0.49188838  44.55447
## 2004.0630       23.25205   8.16647812  38.33762    0.18065976  46.32344
## 2004.0658       22.64902   7.21331587  38.08472   -0.95785150  46.25589
## 2004.0685       21.70167   5.92360718  37.47974   -2.42879711  45.83215
## 2004.0712       24.45815   8.34499435  40.57131   -0.18479688  49.10110
## 2004.0740       27.08434  10.64291370  43.52576    1.93935008  52.22932
## 2004.0767       24.61971   7.85644823  41.38297   -1.01748555  50.25690
## 2004.0795       22.56057   5.48153632  39.63960   -3.55955775  48.68070
## 2004.0822       26.20633   8.81725384  43.59540   -0.38796550  52.80062
## 2004.0849       27.16862   9.47493726  44.86230    0.10846811  54.22877
## 2004.0877       27.43497   9.44183281  45.42810   -0.08315672  54.95309
## 2004.0904       26.50902   8.22133846  44.79670   -1.45957610  54.47762
## 2004.0932       24.08115   5.50358572  42.65871   -4.33078197  52.49308
## 2004.0959       21.82315   2.96015892  40.68614   -7.02530396  50.67160
## 2004.0986       22.29209   3.14793227  41.43625   -6.98637334  51.57056
## 2004.1014       22.35538   2.93412259  41.77664   -7.34687111  52.05764
## 2004.1041       25.56391   5.86944691  45.25837   -4.55617118  55.68399
## 2004.1068       26.88179   6.91786653  46.84572   -3.65039699  57.41398
## 2004.1096       24.08666   3.85686493  44.31646   -6.85214412  55.02547
## 2004.1123       24.97278   4.48055506  45.46501   -6.36737359  56.31293
## 2004.1151       27.03969   6.28836116  47.79103   -4.69673043  58.77612
## 2004.1178       26.98740   5.98015473  47.99464   -5.14040813  59.11520
## 2004.1205       24.71339   3.45331218  45.97346   -7.80109138  57.22786
## 2004.1233       24.44516   2.93523097  45.95510   -8.45144022  57.34177
## 2004.1260       24.73349   2.97656894  46.49042   -8.54085100  58.00784
## 2004.1288       23.06029   1.05914938  45.06143  -10.58755157  56.70813
## 2004.1315       24.47462   2.23194310  46.71730   -9.54261950  58.49186
## 2004.1342       22.49757   0.01594875  44.97919  -11.88510186  56.88024
## 2004.1370       24.51444   1.79639514  47.23249  -10.22981319  59.25870
## 2004.1397       25.71179   2.75974791  48.66383   -9.39032895  60.81391
## 2004.1425       27.11878   3.93510745  50.30245   -8.33758781  62.57515
## 2004.1452       28.42195   5.00893193  51.83496   -7.38516869  64.22906
## 2004.1479       29.21361   5.57347627  52.85373   -6.94085197  65.36806
## 2004.1507       28.66119   4.79610575  52.52627   -7.83730601  65.15968
## 2004.1534       26.46251   2.37457086  50.55044  -10.37681234  63.30183
## 2004.1562       25.81203   1.50328440  50.12078  -11.36498879  62.98905
## 2004.1589       26.59567   2.06809795  51.12324  -10.91601295  64.10735
## 2004.1616       27.09325   2.34879293  51.83771  -10.75013134  64.93663
## 2004.1644       25.46691   0.50745014  50.42637  -12.70528986  63.63911
## 2004.1671       28.32860   3.15597258  53.50123  -10.16961107  66.82681
## 2004.1699       28.63396   3.24995559  54.01796  -10.18752411  67.45544
## 2004.1726       27.89432   2.30068452  53.48795  -11.24776712  67.03640
## 2004.1753       28.70350   2.90193964  54.50506  -10.75658235  68.16359
## 2004.1781       26.77808   0.77025384  52.78591  -12.99745854  66.55362
## 2004.1808       23.92485  -2.28762375  50.13732  -16.16366733  64.01336
## 2004.1836       24.61371  -1.80182026  51.02923  -15.78535581  65.01277
## 2004.1863       25.29568  -1.32135257  51.91272  -15.41156008  66.00293
## 2004.1890       26.98864   0.17160973  53.80567  -14.02446821  68.00175
## 2004.1918       28.28190   1.26635503  55.29744  -13.03480961  69.59861
## 2004.1945       27.40945   0.19683862  54.62206  -14.20864614  69.02754
## 2004.1973       27.22046  -0.18779390  54.62872  -14.69684874  69.13777
## 2004.2000       27.09137  -0.51114390  54.69389  -15.12303473  69.30578
## 2004.2027       28.24861   0.45318248  56.04403  -14.26082563  70.75804
## 2004.2055       29.24106   1.25405833  57.22805  -13.56136323  72.04347
## 2004.2082       28.81009   0.63281800  56.98736  -14.28332752  71.90350
## 2004.2110       27.27129  -1.09497848  55.63755  -16.11117234  70.65375
## 2004.2137       27.38372  -1.17028859  55.93773  -16.28586861  71.05331
## 2004.2164       28.66551  -0.07502305  57.40603  -15.28934001  72.62035
## 2004.2192       28.20447  -0.72137501  57.13031  -16.03379225  72.44273
## 2004.2219       28.48983  -0.62015109  57.59981  -16.03004411  73.00970
## 2004.2247       29.36313   0.07017237  58.65609  -15.43658371  74.16285
## 2004.2274       28.66027  -0.81453250  58.13507  -16.41755033  73.73809
## 2004.2301       30.32187   0.66633877  59.97740  -15.03235055  75.67609
## 2004.2329       29.89718   0.06201610  59.73234  -15.73176520  75.52612
## 2004.2356       30.65006   0.63634031  60.66378  -15.25196384  76.55209
## 2004.2384       31.04096   0.84973810  61.23218  -15.13252988  77.21445
## 2004.2411       30.30644  -0.06125188  60.67412  -16.13693448  76.74981
## 2004.2438       30.78623   0.24310090  61.32937  -15.92545662  77.49792
## 2004.2466       30.04913  -0.66844845  60.76670  -16.92935043  77.02760
## 2004.2493       31.30543   0.41439567  62.19646  -15.93832932  78.54919
## 2004.2521       33.08384   2.02032051  64.14736  -14.42371475  80.59140
## 2004.2548       32.61044   1.37538078  63.84550  -15.15946053  80.38034
## 2004.2575       31.71915   0.31349493  63.12481  -16.31165645  79.74996
## 2004.2603       31.39872  -0.17661327  62.97406  -16.89158678  79.68903
## 2004.2630       32.58777   0.84366254  64.33188  -15.96065299  81.13619
## 2004.2658       32.74668   0.83468894  64.65866  -16.05849612  81.55185
## 2004.2685       33.30654   1.22755292  65.38553  -15.75403660  82.36711
## 2004.2712       31.93427  -0.31085389  64.17939  -17.38039003  81.24892
## 2004.2740       32.34723  -0.06317494  64.75763  -17.22020689  81.91467
## 2004.2767       31.07633  -1.49851740  63.65118  -18.74260121  80.89527
## 2004.2795       31.70973  -1.02873578  64.44820  -18.35943420  81.77890
## 2004.2822       33.52745   0.62617567  66.42872  -16.79070662  83.84561
## 2004.2849       33.81631   0.75302938  66.87958  -16.74961241  84.38223
## 2004.2877       34.71854   1.49404422  67.94303  -16.09393892  85.53101
## 2004.2904       33.56265   0.17772159  66.94757  -17.49519079  84.62049
## 2004.2932       32.95516  -0.58942907  66.49976  -18.34686451  84.25719
## 2004.2959       34.08581   0.38230611  67.78932  -17.45925196  85.63088
## 2004.2986       33.28464  -0.57702948  67.14631  -18.50231542  85.07160
## 2004.3014       32.72647  -1.29263462  66.74557  -19.30125914  84.75419
## 2004.3041       34.08326  -0.09254596  68.25907  -18.18412516  86.35065
## 2004.3068       34.56281   0.23101461  68.89461  -17.94314064  87.06876
## 2004.3096       35.08883   0.60174586  69.57591  -17.65461195  87.83227
## 2004.3123       34.57873  -0.06293523  69.22040  -18.40112711  87.55859
## 2004.3151       33.94761  -0.84795754  68.74318  -19.26761994  87.16285
## 2004.3178       34.60379  -0.34500234  69.55259  -18.84577649  88.05336
## 2004.3205       34.34280  -0.75854870  69.44415  -19.34008053  88.02568
## 2004.3233       34.68704  -0.56620742  69.94028  -19.22814747  88.60222
## 2004.3260       35.33943  -0.06505629  70.74392  -18.80705958  89.48592
## 2004.3288       33.25109  -2.30399878  68.80617  -21.12572475  87.62790
## 2004.3315       33.65895  -2.04610098  69.36400  -20.94721337  88.26511
## 2004.3342       34.75022  -1.10416680  70.60461  -20.08433358  89.58478
## 2004.3370       35.56147  -0.44163825  71.56457  -19.50053150  90.62347
## 2004.3397       33.69237  -2.45884581  69.84358  -21.59614168  88.98087
## 2004.3425       33.32458  -2.97413275  69.62330  -22.18951135  88.83867
## 2004.3452       33.92187  -2.52374875  70.36749  -21.81689405  89.66063
## 2004.3479       34.42248  -2.16945577  71.01441  -21.54005558  90.38501
## 2004.3507       36.00486  -0.73281106  72.74252  -20.18055689  92.19027
## 2004.3534       35.44893  -1.43388965  72.33176  -20.95847669  91.85634
## 2004.3562       36.04645  -0.98095997  73.07386  -20.58208699  92.67499
## 2004.3589       36.30524  -0.86619174  73.47668  -20.54356101  93.15405
## 2004.3616       35.72844  -1.58646020  73.04335  -21.33977746  92.79667
## 2004.3644       35.00293  -2.45489653  72.46075  -22.28387088  92.28973
## 2004.3671       35.79269  -1.80751286  73.39289  -21.71185673  93.29723
## 2004.3699       35.98889  -1.75314559  73.73093  -21.73257467  93.71036
## 2004.3726       35.91186  -1.97148441  73.79521  -22.02571757  93.84945
## 2004.3753       37.14052  -0.88360921  75.16465  -21.01236845  95.29341
## 2004.3781       35.88988  -2.27451795  74.05427  -22.47752837  94.25728
## 2004.3808       36.51058  -1.79356778  74.81472  -22.07055747  95.09171
## 2004.3836       37.04571  -1.39768214  75.48909  -21.74838218  95.83979
## 2004.3863       37.03063  -1.55149432  75.61276  -21.97563869  96.03690
## 2004.3890       36.23782  -2.48255205  74.95819  -22.97987760  95.45551
## 2004.3918       34.78053  -4.07759340  73.63865  -24.64783977  94.20889
## 2004.3945       35.16981  -3.82557248  74.16520  -24.46848208  94.80811
## 2004.3973       35.17821  -3.95395367  74.31038  -24.66927162  95.02570
## 2004.4000       36.17215  -3.09632803  75.44062  -23.88380212  96.22809
## 2004.4027       35.34017  -4.06414007  74.74448  -24.92352070  95.60386
## 2004.4055       37.02464  -2.51503685  76.56432  -23.44607699  97.49536
## 2004.4082       37.58937  -2.08520812  77.26396  -23.08766327  98.26641
## 2004.4110       37.88535  -1.92367769  77.69438  -22.99730585  98.76801
## 2004.4137       37.89229  -2.05073594  77.83532  -23.19529753  98.97988
## 2004.4164       38.08892  -1.98765823  78.16549  -23.20291609  99.38075
## 2004.4192       36.79556  -3.41412477  77.00524  -24.69984409  98.29096
## 2004.4219       37.00514  -3.33721067  77.34748  -24.69315898  98.70343
## 2004.4247       35.84085  -4.63372993  76.31543  -26.05967704  97.74137
## 2004.4274       35.85676  -4.74961724  76.46314  -26.24533520  97.95886
## 2004.4301       36.22227  -4.51547689  76.96003  -26.08073998  98.52529
## 2004.4329       36.00510  -4.86360488  76.87380  -26.49818953  98.50839
## 2004.4356       37.19863  -3.80060394  78.19787  -25.50428875  99.90155
## 2004.4384       36.94062  -4.18873697  78.06997  -25.96130263  99.84254
## 2004.4411       37.36744  -3.89162804  78.62650  -25.73285733 100.46773
## 2004.4438       36.09036  -5.29800451  77.47873  -27.20768223  99.38841
## 2004.4466       35.88632  -5.63094128  77.40359  -27.60885426  99.38150
## 2004.4493       36.72876  -4.91701125  78.37452  -26.96294829 100.42046
## 2004.4521       37.34582  -4.42805647  79.11969  -26.54180833 101.23344
## 2004.4548       36.14395  -5.75763480  78.04554  -27.93899414 100.22690
## 2004.4575       35.84355  -6.18536232  77.87246  -28.43412372 100.12122
## 2004.4603       36.94056  -5.21528971  79.09641  -27.53124958 101.41237
## 2004.4630       36.07806  -6.20435075  78.36047  -28.58730735 100.74343
## 2004.4658       36.46989  -5.93870248  78.87848  -28.38845586 101.32824
## 2004.4685       37.07900  -5.45539966  79.61340  -27.97175168 102.12976
## 2004.4712       37.10515  -5.55469317  79.76498  -28.13744742 102.34774
## 2004.4740       37.39795  -5.38695621  80.18286  -28.03591801 102.83182
## 2004.4767       36.95492  -5.95468683  79.86454  -28.66966320 102.57951
## 2004.4795       37.27445  -5.75950157  80.30841  -28.54030123 103.08921
## 2004.4822       38.26726  -4.89068068  81.42520  -27.73711397 104.27163
## 2004.4849       37.79162  -5.48994894  81.07319  -28.40182785 103.98507
## 2004.4877       37.15979  -6.24505482  80.56464  -29.22219295 103.54178
## 2004.4904       37.91082  -5.61695992  81.43859  -28.65917244 104.48080
## 2004.4932       38.99040  -4.65996190  82.64075  -27.76706555 105.74786
## 2004.4959       39.05072  -4.72187804  82.82332  -27.89369109 105.99513
## 2004.4986       37.39602  -6.49847172  81.29052  -29.73481398 104.52686
## 2004.5014       37.62844  -6.38761682  81.64450  -29.68830956 104.94519
## 2004.5041       38.27069  -5.86658796  82.40798  -29.23145397 105.77284
## 2004.5068       37.46640  -6.79177409  81.72458  -30.22063759 105.15344
## 2004.5096       37.65882  -6.71991722  82.03757  -30.21260387 105.53025
## 2004.5123       38.54743  -5.95154646  83.04641  -29.50788334 106.60275
## 2004.5151       38.58190  -6.03699540  83.20079  -29.65681098 106.82061
## 2004.5178       38.66628  -6.07220177  83.40477  -29.75532592 107.08790
## 2004.5205       38.10379  -6.75397437  82.96155  -30.50023829 106.70781
## 2004.5233       38.99478  -5.98193298  83.97150  -29.79116925 107.78074
## 2004.5260       38.48007  -6.61529662  83.57543  -30.48733911 107.44747
## 2004.5288       37.93729  -7.27640664  83.15098  -31.21109055 107.08567
## 2004.5315       37.79885  -7.53286652  83.13057  -31.53002834 107.12773
## 2004.5342       37.30115  -8.14828932  82.75058  -32.20776679 106.81006
## 2004.5370       38.17304  -7.39380719  83.73989  -31.51543935 107.86152
## 2004.5397       38.72999  -6.95396483  84.41395  -31.13759191 108.59758
## 2004.5425       38.74992  -7.05085190  84.55069  -31.29631539 108.79615
## 2004.5452       37.73859  -8.17869735  83.65587  -32.48583994 107.96302
## 2004.5479       38.25482  -7.77868908  84.28832  -32.14735466 108.65699
## 2004.5507       37.44424  -8.70519666  83.59367  -33.13523030 108.02370
## 2004.5534       37.38363  -8.88143373  83.64870  -33.37268165 108.13995
## 2004.5562       37.20438  -9.17604005  83.58479  -33.72834964 108.13710
## 2004.5589       38.37489  -8.12058896  84.87037  -32.73380872 109.48359
## 2004.5616       38.50570  -8.10455491  85.11596  -32.77853448 109.78994
## 2004.5644       37.57373  -9.15102020  84.29849  -33.88561034 109.03308
## 2004.5671       38.29604  -8.54292649  85.13501  -33.33797903 109.93006
## 2004.5699       38.59821  -8.35469995  85.55112  -33.21006781 110.40648
## 2004.5726       38.57277  -8.49380083  85.63934  -33.40933801 110.55488
## 2004.5753       38.95516  -8.22480052  86.13512  -33.20036206 111.11068
## 2004.5781       38.77285  -8.52022591  86.06593  -33.55566789 111.10137
## 2004.5808       37.47278  -9.93313882  84.87871  -35.02831836 109.97389
## 2004.5836       37.80941  -9.70909091  85.32791  -34.86386615 110.48269
## 2004.5863       38.07499  -9.55582133  85.70581  -34.77005142 110.92004
## 2004.5890       38.72859  -9.01427092  86.47146  -34.28781598 111.74500
## 2004.5918       38.32938  -9.52526736  86.18403  -34.85798851 111.51675
## 2004.5945       38.52241  -9.44376743  86.48858  -34.83552677 111.88034
## 2004.5973       38.24781  -9.82963628  86.32525  -35.28029685 111.77591
## 2004.6000       38.93087  -9.25758224  87.11932  -34.76700804 112.62875
## 2004.6027       37.08008 -11.21912316  85.37929  -36.78717913 110.94735
## 2004.6055       35.98888 -12.42082428  84.39859  -38.04737627 110.02515
## 2004.6082       35.53528 -12.98467677  84.05524  -38.66959156 109.74016
## 2004.6110       35.90615 -12.72380444  84.53611  -38.46694972 110.27926
## 2004.6137       37.25580 -11.48390901  85.99551  -37.28515337 111.79676
## 2004.6164       37.52158 -11.32763107  86.37080  -37.18684397 112.23001
## 2004.6192       36.92394 -12.03453486  85.88242  -37.95158664 111.79947
## 2004.6219       35.27526 -13.79223172  84.34275  -39.76699359 110.31752
## 2004.6247       34.93507 -14.24120243  84.11133  -40.27354645 110.14368
## 2004.6274       36.29109 -12.99371777  85.57589  -39.08351685 111.66569
## 2004.6301       36.40665 -12.98645364  85.79975  -39.13358154 111.94687
## 2004.6329       36.14076 -13.36040468  85.64192  -39.56473598 111.84625
## 2004.6356       36.89777 -12.71121810  86.50675  -38.97262819 112.76816
## 2004.6384       36.57974 -13.13683574  86.29631  -39.45520083 112.61468
## 2004.6411       37.63636 -12.18757256  87.46029  -38.56276966 113.83549
## 2004.6438       37.81953 -12.11153153  87.75059  -38.54343845 114.18250
## 2004.6466       36.71302 -13.32493640  86.75098  -39.81343172 113.23948
## 2004.6493       36.32287 -13.82175878  86.46750  -40.36672188 113.01246
## 2004.6521       36.04087 -14.21020205  86.29194  -40.81151305 112.89325
## 2004.6548       34.90720 -15.45009209  85.26449  -42.10763188 111.92203
## 2004.6575       35.29591 -15.16737464  85.75920  -41.88102488 112.47285
## 2004.6603       34.90530 -15.66376377  85.47436  -42.43340684 112.24400
## 2004.6630       34.53673 -16.13787819  85.21135  -42.96339723 112.03686
## 2004.6658       34.72782 -16.05212940  85.50776  -42.93340826 112.38904
## 2004.6685       35.39459 -15.49047161  86.27965  -42.42739486 113.21657
## 2004.6712       35.64422 -15.34573310  86.63418  -42.33818602 113.62663
## 2004.6740       35.40171 -15.69293001  86.49635  -42.74079862 113.54422
## 2004.6767       34.11592 -17.08319198  85.31503  -44.18636296 112.41820
## 2004.6795       35.10043 -16.20293324  86.40380  -43.36129399 113.56216
## 2004.6822       35.30470 -16.10271174  86.71211  -43.31615033 113.92555
## 2004.6849       35.36310 -16.14814788  86.87434  -43.41655305 114.14274
## 2004.6877       33.27493 -18.33993446  84.88980  -45.66319565 112.21306
## 2004.6904       33.36470 -18.35359044  85.08298  -45.73159773 112.46099
## 2004.6932       32.42228 -19.39921355  84.24378  -46.83185769 111.67643
## 2004.6959       32.80239 -19.12211780  84.72689  -46.60929018 112.21406
## 2004.6986       31.29048 -20.73682986  83.31778  -48.27842253 110.85938
## 2004.7014       32.10386 -20.02604528  84.23377  -47.62195091 111.82967
## 2004.7041       32.82214 -19.41015996  85.05445  -47.06027188 112.70456
## 2004.7068       32.79602 -19.53848217  85.13052  -47.24269431 112.83473
## 2004.7096       32.64115 -19.79534975  85.07765  -47.55355667 112.83586
## 2004.7123       31.83624 -20.70205787  84.37455  -48.51415474 112.18664
## 2004.7151       32.26565 -20.37425987  84.90555  -48.24014248 112.77143
## 2004.7178       33.52658 -19.21473105  86.26790  -47.13429578 114.18746
## 2004.7205       33.56537 -19.27715676  86.40790  -47.25030060 114.38104
## 2004.7233       33.47551 -19.46803223  86.41906  -47.49465275 114.44568
## 2004.7260       33.41300 -19.63137054  86.45738  -47.71136589 114.53737
## 2004.7288       30.62072 -22.52429376  83.76573  -50.65756267 111.89900
## 2004.7315       30.19323 -23.05222157  83.43869  -51.23866336 111.62513
## 2004.7342       30.18195 -23.16376725  83.52766  -51.40328180 111.76717
## 2004.7370       29.55750 -23.88828212  83.00328  -52.18076987 111.29577
## 2004.7397       31.34930 -22.19636514  84.89496  -50.54172709 113.24032
## 2004.7425       32.07855 -21.56680894  85.72391  -49.96494665 114.12205
## 2004.7452       33.08041 -20.66445547  86.82528  -49.11527104 115.27610
## 2004.7479       32.90438 -20.93982056  86.74857  -49.44321664 115.25197
## 2004.7507       32.97420 -20.96914269  86.91754  -49.52502246 115.47342
## 2004.7534       32.59299 -21.44931250  86.63529  -50.05757967 115.24356
## 2004.7562       30.89193 -23.24915597  85.03301  -51.90971478 113.69357
## 2004.7589       29.06668 -25.17300581  83.30637  -53.88576103 112.01912
## 2004.7616       27.82460 -26.51350321  82.16271  -55.27836014 110.92757
## 2004.7644       29.49235 -24.94400385  83.92870  -53.76086827 112.74557
## 2004.7671       30.61111 -23.92331082  85.14553  -52.79208904 114.01431
## 2004.7699       31.43556 -23.19674980  86.06787  -52.11734864 114.98847
## 2004.7726       32.41830 -22.31173114  87.14832  -51.28405791 116.12065
## 2004.7753       32.49348 -22.33408555  87.32106  -51.35804806 116.34502
## 2004.7781       32.00232 -22.92261804  86.92726  -51.99812459 116.00277
## 2004.7808       30.64694 -24.37519545  85.66908  -53.50215481 114.79604
## 2004.7836       30.68177 -24.43739487  85.80093  -53.61571633 114.97925
## 2004.7863       30.74598 -24.47003372  85.96200  -53.69962702 115.19159
## 2004.7890       28.20937 -27.10332850  83.52207  -56.38410386 112.80285
## 2004.7918       27.97080 -27.43841926  83.38002  -56.77028736 112.71188
## 2004.7945       28.74415 -26.76141587  84.24972  -56.14428788 113.63259
## 2004.7973       28.98285 -26.61890078  84.58460  -56.05268832 114.01838
## 2004.8000       30.53876 -25.15900756  86.23652  -54.64362270 115.72114
## 2004.8027       31.08754 -24.70607308  86.88116  -54.24142835 116.41651
## 2004.8055       30.40679 -25.48251359  86.29609  -55.06852198 115.88210
## 2004.8082       29.89525 -26.08956952  85.88008  -55.72614445 115.51665
## 2004.8110       29.66012 -26.42005876  85.74031  -56.10711410 115.42736
## 2004.8137       30.53320 -25.64217808  86.70858  -55.37962814 116.44603
## 2004.8164       29.17009 -27.10032435  85.44051  -56.88808387 115.22827
## 2004.8192       28.24376 -28.12152954  84.60906  -57.95951369 114.44704
## 2004.8219       27.29625 -29.16376314  83.75626  -59.05188753 113.64438
## 2004.8247       28.10239 -28.45218213  84.65696  -58.39036278 114.59514
## 2004.8274       28.14285 -28.50611890  84.79182  -58.49427226 114.77997
## 2004.8301       29.10980 -27.63341043  85.85302  -57.67145336 115.89106
## 2004.8329       26.81817 -30.01913058  83.65547  -60.10698036 113.74332
## 2004.8356       25.08926 -31.84197277  82.02049  -61.97954708 112.15807
## 2004.8384       27.16000 -29.86501319  84.18501  -60.05223012 114.37222
## 2004.8411       28.63714 -28.48149143  85.75578  -58.71826949 115.99255
## 2004.8438       27.33552 -29.87658442  84.54762  -60.16284250 114.83388
## 2004.8466       24.49701 -32.80840742  81.80243  -63.14406482 112.13809
## 2004.8493       25.26265 -32.13593319  82.66124  -62.52090959 113.04622
## 2004.8521       25.11165 -32.37994664  82.60326  -62.81416213 113.03747
## 2004.8548       25.35216 -32.23230636  82.93663  -62.71568139 113.42000
## 2004.8575       26.19308 -31.48409793  83.87026  -62.01655336 114.40272
## 2004.8603       29.20590 -28.56384610  86.97565  -59.14530316 117.55711
## 2004.8630       26.40150 -31.46066302  84.26367  -62.09104332 114.89405
## 2004.8658       24.21487 -33.73956376  82.16931  -64.41878928 112.84853
## 2004.8685       25.53978 -32.50677995  83.58634  -63.23477304 114.31433
## 2004.8712       26.85189 -31.28664751  84.99043  -62.06333091 115.76711
## 2004.8740       24.90127 -33.32910458  83.13164  -64.15440136 113.95694
## 2004.8767       23.90887 -34.41319213  82.23093  -65.28702576 113.10476
## 2004.8795       23.80878 -34.60482829  82.22238  -65.52712258 113.14467
## 2004.8822       25.05204 -33.45297009  83.55704  -64.42364922 114.52772
## 2004.8849       23.43831 -35.15795102  82.03458  -66.17693950 113.05357
## 2004.8877       23.76433 -34.92305105  82.45171  -65.99027377 113.51893
## 2004.8904       25.34406 -33.43430026  84.12241  -64.54968245 115.23779
## 2004.8932       26.28641 -32.58278135  85.15560  -63.74624859 116.31907
## 2004.8959       24.99060 -33.96928433  83.95049  -65.18076253 115.16197
## 2004.8986       25.35943 -33.69100677  84.40988  -64.95042218 115.66929
## 2004.9014       26.16791 -32.97294757  85.30877  -64.28022681 116.61605
## 2004.9041       24.76349 -34.46764579  83.99463  -65.82271579 115.34970
## 2004.9068       25.16508 -34.15619496  84.48636  -65.55898299 115.88915
## 2004.9096       27.95376 -31.45752167  87.36505  -62.90795532 118.81548
## 2004.9123       25.43331 -34.06784597  84.93446  -65.56585318 116.43247
## 2004.9151       23.22653 -36.36435111  82.81742  -67.90986013 114.36293
## 2004.9178       23.51102 -36.16946267  83.19150  -67.76240208 114.78444
## 2004.9205       22.96464 -36.80530213  82.73459  -68.44560083 114.37489
## 2004.9233       24.85602 -35.00325271  84.71530  -66.69083992 116.40289
## 2004.9260       23.18303 -36.76543917  83.13151  -68.50024442 114.86631
## 2004.9288       23.64472 -36.39282001  83.68226  -68.17477315 115.46421
## 2004.9315       20.78873 -39.33774558  80.91520  -71.16677677 112.74423
## 2004.9342       22.33401 -37.88126691  82.54928  -69.75730662 114.42532
## 2004.9370       22.45518 -37.84875956  82.75913  -69.77173857 114.68211
## 2004.9397       23.90995 -36.48253323  84.30243  -68.45238262 116.27228
## 2004.9425       23.64067 -36.84022311  84.12156  -68.85687426 116.13822
## 2004.9452       21.31118 -39.25799821  81.88035  -71.32138281 113.94374
## 2004.9479       21.61086 -39.04646952  82.26819  -71.15651956 114.37824
## 2004.9507       21.30156 -39.44379475  82.04691  -71.60044250 114.20356
## 2004.9534       20.28882 -40.54443294  81.12207  -72.74761097 113.32525
## 2004.9562       20.95629 -39.96473635  81.87731  -72.21437754 114.12695
## 2004.9589       21.13757 -39.87109998  82.14623  -72.16713748 114.44227
## 2004.9616       20.04842 -41.04776816  81.14460  -73.39013540 113.48697
## 2004.9644       18.73040 -42.45317600  79.91398  -74.84180673 112.30261
## 2004.9671       16.53958 -44.73127370  77.81042  -77.16610192 110.24525
## 2004.9699       17.33594 -44.02204940  78.69394  -76.50300940 111.17490
## 2004.9726       17.47624 -43.96877830  78.92125  -76.49580466 111.44828
## 2004.9753       20.61486 -40.91705770  82.14677  -73.49008527 114.71980
## 2004.9781       21.01791 -40.60078415  82.63660  -73.21974805 115.25556
## 2004.9808       21.36981 -40.33553286  83.07515  -73.00036850 115.73999
## 2004.9836       22.84619 -38.94568864  84.63806  -71.65633169 117.34871
## 2004.9863       20.07386 -41.80442468  81.95215  -74.56081108 114.70854
## 2004.9890       21.32119 -40.64339120  83.28576  -73.44545716 116.08783
## 2004.9918       21.08895 -40.96179715  83.13970  -73.80947914 115.98738
## 2004.9945       22.44464 -39.69215862  84.58144  -72.58539338 117.47468
## 2004.9973       23.88663 -38.33610286  86.10936  -71.27482740 119.04809
## 2005.0000       24.41530 -37.89324283  86.72385  -70.87739441 119.70800
## 2005.0027       23.64690 -38.74733864  86.04114  -71.77685479 119.07066
## 2005.0055       22.56934 -39.91048436  85.04915  -72.98530285 118.12397
## 2005.0082       21.33776 -41.22752399  83.90304  -74.34758286 117.02310
## 2005.0110       22.07243 -40.57819189  84.72306  -73.74342942 117.88830
## 2005.0137       20.91589 -41.81996321  83.65174  -75.03031794 116.86210
## 2005.0164       24.55218 -38.26878530  87.37315  -71.52419603 120.62856
## 2005.0192       21.65143 -41.25453362  84.55739  -74.55493938 117.85780
## 2005.0219       19.43034 -43.56050666  82.42119  -76.90584674 115.76653
## 2005.0247       24.41696 -38.65865289  87.49258  -72.04886682 120.88279
## 2005.0274       25.88847 -37.27180161  89.04874  -70.70682916 122.48377
## 2005.0301       26.02004 -37.22477222  89.26485  -70.70455341 122.74463
## 2005.0329       24.37379 -38.95544773  87.70303  -72.47992282 121.22751
## 2005.0356       22.24627 -41.16728469  85.65983  -74.73639417 119.22894
## 2005.0384       23.14929 -40.34847381  86.64705  -73.96215840 120.26073
## 2005.0411       22.54337 -41.03848908  86.12522  -74.69668976 119.78342
## 2005.0438       21.98502 -41.68081318  85.65086  -75.38347115 119.35352
## 2005.0466       22.87562 -40.87408619  86.62533  -74.62114289 120.37239
## 2005.0493       21.75903 -42.07443916  85.59250  -75.86583624 119.38390
## 2005.0521       20.15676 -43.76035580  84.07388  -77.59603516 117.90956
## 2005.0548       23.04116 -40.95950324  87.04182  -74.83940700 120.92172
## 2005.0575       24.01153 -40.07256325  88.09563  -73.99663376 122.01970
## 2005.0603       22.03129 -42.13612809  86.19871  -76.10430792 120.16689
## 2005.0630       23.25205 -40.99858747  87.50268  -75.01081942 121.51492
## 2005.0658       22.64902 -41.68472592  86.98276  -75.74095301 121.03899
## 2005.0685       21.70167 -42.71507149  86.11842  -76.81523696 120.21859
## 2005.0712       24.45815 -40.04148715  88.95779  -74.18553445 123.10184
## 2005.0740       27.08434 -37.49809231  91.66677  -71.68596512 125.85464
## 2005.0767       24.61971 -40.04540323  89.28482  -74.27704544 123.51646
## 2005.0795       22.56057 -42.18711863  87.30826  -76.46247435 121.58361
## 2005.0822       26.20633 -38.62383269  91.03649  -72.94284624 125.35550
## 2005.0849       27.16862 -37.74390796  92.08115  -72.10652386 126.44376
## 2005.0877       27.43497 -37.55982272  92.42976  -71.96598573 126.83592
## 2005.0904       26.50902 -38.56792613  91.58597  -73.01758120 126.03562
## 2005.0932       24.08115 -41.07785387  89.24015  -75.57094615 123.73324
## 2005.0959       21.82315 -43.41780670  87.06410  -77.95428157 121.60058
## 2005.0986       22.29209 -43.03071160  87.61489  -77.61051463 122.19470
## 2005.1014       22.35538 -43.04916730  87.75993  -77.67224428 122.38301
## 2005.1041       25.56391 -39.92228538  91.05010  -74.58858227 125.71640
## 2005.1068       26.88179 -38.68594484  92.44953  -73.39540783 127.15899
## 2005.1096       24.08666 -41.56251321  89.73584  -76.31508869 124.48842
## 2005.1123       24.97278 -40.75773823  90.70330  -75.55337278 125.49893
## 2005.1151       27.03969 -38.77206516  92.85145  -73.61070556 127.69009
## 2005.1178       26.98740 -38.90550008  92.88030  -73.78709331 127.76189
## 2005.1205       24.71339 -41.26055156  90.68732  -76.18504479 125.61182
## 2005.1233       24.44516 -41.60971389  90.50004  -76.57705448 125.46738
## 2005.1260       24.73349 -41.40222721  90.86921  -76.41236273 125.87935
## 2005.1288       23.06029 -43.15617195  89.27675  -78.20905016 124.32963
## 2005.1315       24.47462 -41.82248636  90.77173  -76.91805520 125.86730
## 2005.1342       22.49757 -43.88008572  88.87522  -79.01829331 124.01343
## 2005.1370       24.51444 -41.94365968  90.97255  -77.12445436 126.15334
## 2005.1397       25.71179 -40.82666531  92.25024  -76.04999558 127.47357
## 2005.1425       27.11878 -39.49992883  93.73749  -74.76574339 129.00330
## 2005.1452       28.42195 -38.27692234  95.12081  -73.58517007 130.42906
## 2005.1479       29.21361 -37.56532456  95.99253  -72.91595453 131.34316
## 2005.1507       28.66119 -38.19770706  95.52008  -73.59066850 130.91305
## 2005.1534       26.46251 -40.47625910  93.40127  -75.91150146 128.83652
## 2005.1562       25.81203 -41.20651050  92.83057  -76.68398339 128.30805
## 2005.1589       26.59567 -40.50255480  93.69389  -76.02220800 129.21354
## 2005.1616       27.09325 -40.08455820  94.27106  -75.64634169 129.83284
## 2005.1644       25.46691 -41.79038980  92.72421  -77.39425372 128.32807
## 2005.1671       28.32860 -39.00809863  95.66530  -74.65399330 131.31119
## 2005.1699       28.63396 -38.78204341  96.04996  -74.46991934 131.73784
## 2005.1726       27.89432 -39.60089480  95.38953  -75.33070265 131.11934
## 2005.1753       28.70350 -38.87083028  96.27783  -74.64252091 132.04952
## 2005.1781       26.77808 -40.87527644  94.43144  -76.68880087 130.24496
## 2005.1808       23.92485 -43.80744525  91.65714  -79.66275466 127.51245
## 2005.1836       24.61371 -43.19742642  92.42484  -79.09447218 128.32189
## 2005.1863       25.29568 -42.59420088  93.18557  -78.53293451 129.12430
## 2005.1890       26.98864 -40.97990361  94.95718  -76.96027682 130.93756
## 2005.1918       28.28190 -39.76521292  96.32901  -75.78717757 132.35097
## 2005.1945       27.40945 -40.71614142  95.53504  -76.77964955 131.59854
## 2005.1973       27.22046 -40.98351260  95.42444  -77.08851640 131.52944
## 2005.2000       27.09137 -41.19089801  95.37365  -77.33734986 131.52010
## 2005.2027       28.24861 -40.11187501  96.60909  -76.29972743 132.79694
## 2005.2055       29.24106 -39.19754277  97.67965  -75.42674844 133.90886
## 2005.2082       28.81009 -39.70654011  97.32672  -75.97705189 133.59723
## 2005.2110       27.27129 -41.32328109  95.86586  -77.63505199 132.17763
## 2005.2137       27.38372 -41.28869816  96.05614  -77.64168135 132.40912
## 2005.2164       28.66551 -40.08467781  97.41569  -76.47882663 133.80984
## 2005.2192       28.20447 -40.62338980  97.03233  -77.05865773 133.46760
## 2005.2219       28.48983 -40.41561804  97.39528  -76.89195873 133.87162
## 2005.2247       29.36313 -39.61981694  98.34608  -76.13718420 134.86345
## 2005.2274       28.66027 -40.40009309  97.72063  -76.95844088 134.27898
## 2005.2301       30.32187 -38.81582143  99.45956  -75.41510386 136.05884
## 2005.2329       29.89718 -39.31775207  99.11211  -75.95792341 135.75228
## 2005.2356       30.65006 -38.64202482  99.94215  -75.32303948 136.62316
## 2005.2384       31.04096 -38.32819418 100.41012  -75.05000674 137.13193
## 2005.2411       30.30644 -39.13970329  99.75257  -75.90226847 136.51514
## 2005.2438       30.78623 -38.73680392 100.30927  -75.54007659 137.11254
## 2005.2466       30.04913 -39.55072374  99.64898  -76.39465894 136.49291
## 2005.2493       31.30543 -38.37115049 100.98201  -75.25570339 137.86656
## 2005.2521       33.08384 -36.66938068 102.83707  -73.59450659 139.76219
## 2005.2548       32.61044 -37.21934385 102.44022  -74.18499824 139.40588
## 2005.2575       31.71915 -38.18710620 101.62541  -75.19324469 138.63155
## 2005.2603       31.39872 -38.58392905 101.38138  -75.63050740 138.42795
## 2005.2630       32.58777 -37.47119155 102.64673  -74.55816565 139.73371
## 2005.2658       32.74668 -37.38851298 102.88186  -74.51583889 140.00919
## 2005.2685       33.30654 -36.90479262 103.51787  -74.07242653 140.68550
## 2005.2712       31.93427 -38.35312546 102.22166  -75.56102371 139.42956
## 2005.2740       32.34723 -38.01614191 102.71060  -75.26426096 139.95872
## 2005.2767       31.07633 -39.36293645 101.51560  -76.65123293 138.80390
## 2005.2795       31.70973 -38.80535122 102.22482  -76.13378187 139.55325
## 2005.2822       33.52745 -37.06336839 104.11827  -74.43189012 141.48679
## 2005.2849       33.81631 -36.85016380 104.48278  -74.25873364 141.89135
## 2005.2877       34.71854 -36.02350709 105.46058  -73.47208222 142.90915
## 2005.2904       33.56265 -37.25488569 104.38018  -74.74342341 141.86872
## 2005.2932       32.95516 -37.93777925 103.84811  -75.46623700 141.37657
## 2005.2959       34.08581 -36.88246324 105.05409  -74.45079860 142.62242
## 2005.2986       33.28464 -37.75888389 104.32817  -75.36705459 141.93634
## 2005.3014       32.72647 -38.39222981 103.84516  -76.04019369 141.49313
## 2005.3041       34.08326 -37.11052775 105.27705  -74.79824280 142.96476
## 2005.3068       34.56281 -36.70598993 105.83161  -74.43341426 143.55904
## 2005.3096       35.08883 -36.25490811 106.43256  -74.02199997 144.19965
## 2005.3123       34.57873 -36.83985608 105.99732  -74.64657386 143.80404
## 2005.3151       33.94761 -37.54575369 105.44098  -75.39205589 143.28728
## 2005.3178       34.60379 -36.96427338 106.17186  -74.85011864 144.05770
## 2005.3205       34.34280 -37.29988558 105.98549  -75.22523268 143.91083
## 2005.3233       34.68704 -37.03019267 106.40427  -74.99500050 144.36907
## 2005.3260       35.33943 -36.45226416 107.13113  -74.45649175 145.13535
## 2005.3288       33.25109 -38.61499546 105.11717  -76.65860197 143.16078
## 2005.3315       33.65895 -38.28144475 105.59935  -76.36438946 143.68229
## 2005.3342       34.75022 -37.26440823 106.76485  -75.38665054 144.88709
## 2005.3370       35.56147 -36.52732032 107.65026  -74.68881976 145.81176
## 2005.3397       33.69237 -38.47050410 105.85524  -76.67122033 144.05595
## 2005.3425       33.32458 -38.91229558 105.56146  -77.15218839 143.80135
## 2005.3452       33.92187 -38.38893733 106.23268  -76.66796662 144.51171
## 2005.3479       34.42248 -37.96218437 106.80714  -76.28031017 145.12527
## 2005.3507       36.00486 -36.45358713 108.46330  -74.81076958 146.82048
## 2005.3534       35.44893 -37.08321395 107.98108  -75.47941333 146.37728
## 2005.3562       36.04645 -36.55932673 108.65223  -74.99450343 147.08740
## 2005.3589       36.30524 -36.37408877 108.98458  -74.84820330 147.45869
## 2005.3616       35.72844 -37.02436902 108.48126  -75.53738202 146.99427
## 2005.3644       35.00293 -37.82329250 107.82915  -76.37516472 146.38102
## 2005.3671       35.79269 -37.10686532 108.69224  -75.69755762 147.28293
## 2005.3699       35.98889 -36.98391792 108.96171  -75.61339130 147.59118
## 2005.3726       35.91186 -37.13413420 108.95786  -75.80234976 147.62608
## 2005.3753       37.14052 -35.97858834 110.25963  -74.68550731 148.96655
## 2005.3781       35.88988 -37.30227272 109.08203  -76.04785643 147.82761
## 2005.3808       36.51058 -36.75453899 109.77569  -75.53874890 148.55990
## 2005.3836       37.04571 -36.29230522 110.38372  -75.11510290 149.20651
## 2005.3863       37.03063 -36.38019940 110.44147  -75.24154653 149.30281
## 2005.3890       36.23782 -37.24576409 109.72140  -76.14562247 148.62126
## 2005.3918       34.78053 -38.77573225 108.33679  -77.71406379 147.27512
## 2005.3945       35.16981 -38.45905301 108.79868  -77.43581973 147.77544
## 2005.3973       35.17821 -38.52318583 108.87961  -77.53834987 147.89478
## 2005.4000       36.17215 -37.60171695 109.94601  -76.65524056 148.99953
## 2005.4027       35.34017 -38.50608614 109.18642  -77.59793168 148.27827
## 2005.4055       37.02464 -36.89393583 110.94321  -76.02406577 150.07334
## 2005.4082       37.58937 -36.40145118 111.58020  -75.56982810 150.74858
## 2005.4110       37.88535 -36.17765153 111.94836  -75.38423811 151.15495
## 2005.4137       37.89229 -36.24282284 112.02741  -75.48758188 151.27217
## 2005.4164       38.08892 -36.11823613 112.29607  -75.40113055 151.57897
## 2005.4192       36.79556 -37.48356735 111.07468  -76.80456015 150.39567
## 2005.4219       37.00514 -37.34588743 111.35616  -76.70494174 150.71521
## 2005.4247       35.84085 -38.58200626 110.26370  -77.97908530 149.66078
## 2005.4274       35.85676 -38.63785446 110.35138  -78.07292158 149.78644
## 2005.4301       36.22227 -38.34403236 110.78858  -77.81705099 150.26160
## 2005.4329       36.00510 -38.63283204 110.64303  -78.14376573 150.15396
## 2005.4356       37.19863 -37.51085239 111.90812  -77.05966479 151.45693
## 2005.4384       36.94062 -37.84035251 111.72159  -77.42700737 151.30824
## 2005.4411       37.36744 -37.48495276 112.21983  -77.10941395 151.84429
## 2005.4438       36.09036 -38.83337682 111.01410  -78.49560829 150.67633
## 2005.4466       35.88632 -39.10869601 110.88135  -78.80866183 150.58131
## 2005.4493       36.72876 -38.33747966 111.79499  -78.07514401 151.53265
## 2005.4521       37.34582 -37.79156636 112.48320  -77.56689350 152.25852
## 2005.4548       36.14395 -39.06451050 111.35241  -78.87746480 151.16537
## 2005.4575       35.84355 -39.43592481 111.12302  -79.28647074 150.97357
## 2005.4603       36.94056 -38.40985663 112.29098  -78.29795877 152.17908
## 2005.4630       36.07806 -39.34323646 111.49936  -79.26885948 151.42498
## 2005.4658       36.46989 -39.02221813 111.96200  -78.98532680 151.92511
## 2005.4685       37.07900 -38.48385322 112.64186  -78.48441241 152.64242
## 2005.4712       37.10515 -38.52838949 112.73868  -78.56636417 152.77665
## 2005.4740       37.39795 -38.30619704 113.10210  -78.38155228 153.17745
## 2005.4767       36.95492 -38.81977091 112.72962  -78.93247188 152.84232
## 2005.4795       37.27445 -38.57072465 113.11963  -78.72073661 153.26964
## 2005.4822       38.26726 -37.64833557 114.18285  -77.83562388 154.37014
## 2005.4849       37.79162 -38.19432555 113.77757  -78.41885566 154.00210
## 2005.4877       37.15979 -38.89644021 113.21602  -79.15817768 153.47776
## 2005.4904       37.91082 -38.21563834 114.03727  -78.51454881 154.33618
## 2005.4932       38.99040 -37.20621482 115.18701  -77.54226404 155.52306
## 2005.4959       39.05072 -37.21598420 115.31742  -77.58913800 155.69058
## 2005.4986       37.39602 -38.94070719 113.73275  -79.35093150 154.14298
## 2005.5014       37.62844 -38.77825499 114.03513  -79.22551583 154.48239
## 2005.5041       38.27069 -38.20589963 114.74729  -78.69016313 155.23155
## 2005.5068       37.46640 -39.08002747 114.01283  -79.60125984 154.53406
## 2005.5096       37.65882 -38.95737799 114.27503  -79.51554552 154.83319
## 2005.5123       38.54743 -38.13847780 115.23334  -78.73354689 155.82841
## 2005.5151       38.58190 -38.17365800 115.33745  -78.80559514 155.96939
## 2005.5178       38.66628 -38.15885391 115.49142  -78.82762568 156.16020
## 2005.5205       38.10379 -38.79087192 114.99844  -79.49644498 155.70402
## 2005.5233       38.99478 -37.96932946 115.95890  -78.71167057 156.70124
## 2005.5260       38.48007 -38.55344319 115.51357  -79.33251920 156.29265
## 2005.5288       37.93729 -39.16555217 115.04013  -79.98133002 155.85591
## 2005.5315       37.79885 -39.37325763 114.97096  -80.22570435 155.82341
## 2005.5342       37.30115 -39.94017036 114.54246  -80.82925306 155.43154
## 2005.5370       38.17304 -39.13742032 115.48350  -80.06310621 156.40919
## 2005.5397       38.72999 -38.64955003 116.10954  -79.61180639 157.07179
## 2005.5425       38.74992 -38.69864699 116.19848  -79.69744121 157.19728
## 2005.5452       37.73859 -39.77893804 115.25611  -80.81423758 156.29141
## 2005.5479       38.25482 -39.33160898 115.84124  -80.40338140 156.91301
## 2005.5507       37.44424 -40.21102733 115.09950  -81.31924027 156.20771
## 2005.5534       37.38363 -40.34040467 115.10767  -81.48502585 156.25230
## 2005.5562       37.20438 -40.58837877 114.99713  -81.76937601 156.17813
## 2005.5589       38.37489 -39.48652097 116.23630  -80.70386216 157.45364
## 2005.5616       38.50570 -39.42430376 116.43571  -80.67795688 157.68936
## 2005.5644       37.57373 -40.42480753 115.57227  -81.71474065 156.86221
## 2005.5671       38.29604 -39.77097200 116.36306  -81.09715327 157.68924
## 2005.5699       38.59821 -39.53722148 116.73364  -80.89961913 158.09603
## 2005.5726       38.57277 -39.63101435 116.77655  -81.02959670 158.17514
## 2005.5753       38.95516 -39.31692016 117.22724  -80.75165561 158.66197
## 2005.5781       38.77285 -39.56746399 117.11316  -81.03832102 158.58402
## 2005.5808       37.47278 -40.93570586 115.88127  -82.44265304 157.38822
## 2005.5836       37.80941 -40.66719568 116.28602  -82.21020165 157.82902
## 2005.5863       38.07499 -40.46967085 116.61966  -82.04870434 158.19869
## 2005.5890       38.72859 -39.88407047 117.34125  -81.49910030 158.95628
## 2005.5918       38.32938 -40.35122053 117.00998  -82.00221558 158.66098
## 2005.5945       38.52241 -40.22607613 117.27089  -81.91300537 158.95782
## 2005.5973       38.24781 -40.56850075 117.06411  -82.29133324 158.78695
## 2005.6000       38.93087 -39.95320108 117.81494  -81.71190595 159.57365
## 2005.6027       37.08008 -41.87169336 116.03186  -83.66623981 157.82641
## 2005.6055       35.98888 -43.03054120 115.00831  -84.86089853 156.83867
## 2005.6082       35.53528 -43.55173422 114.62230  -85.41787179 156.48844
## 2005.6110       35.90615 -43.24839465 115.06070  -85.15028192 156.96259
## 2005.6137       37.25580 -41.96622267 116.47783  -83.90382916 158.41543
## 2005.6164       37.52158 -41.76785735 116.81103  -83.74115266 158.78432
## 2005.6192       36.92394 -42.43286141 116.28074  -84.44181522 158.28970
## 2005.6219       35.27526 -44.14884470 114.69937  -86.19342677 156.74395
## 2005.6247       34.93507 -44.55628653 114.42642  -86.63646669 156.50660
## 2005.6274       36.29109 -43.26745623 115.84963  -85.38320440 157.96538
## 2005.6301       36.40665 -43.21902825 116.03232  -85.37031441 158.18361
## 2005.6329       36.14076 -43.55199582 115.83351  -85.73879004 158.02030
## 2005.6356       36.89777 -42.86200474 116.65754  -85.08427714 158.87981
## 2005.6384       36.57974 -43.24699545 116.40647  -85.50471626 158.66420
## 2005.6411       37.63636 -42.25728155 117.53000  -84.55042107 159.82314
## 2005.6438       37.81953 -42.14096465 117.78002  -84.46949322 160.10855
## 2005.6466       36.71302 -43.31426715 116.74031  -85.67815522 159.10420
## 2005.6493       36.32287 -43.77115935 116.41690  -86.17037743 158.81612
## 2005.6521       36.04087 -44.11984330 116.20158  -86.55436197 158.63610
## 2005.6548       34.90720 -45.32014359 115.13454  -87.78993351 157.60433
## 2005.6575       35.29591 -44.99800468 115.58983  -87.50303658 158.09486
## 2005.6603       34.90530 -45.45513936 115.26573  -87.99538405 157.80598
## 2005.6630       34.53673 -45.89016511 114.96363  -88.46559346 157.53906
## 2005.6658       34.72782 -45.76549216 115.22112  -88.37607513 157.83171
## 2005.6685       35.39459 -45.16507351 115.95425  -87.81078211 158.59996
## 2005.6712       35.64422 -44.98173622 116.27018  -87.66254154 158.95099
## 2005.6740       35.40171 -45.29049524 116.09392  -88.00636845 158.80979
## 2005.6767       34.11592 -46.64247901 114.87431  -89.39339133 157.62522
## 2005.6795       35.10043 -45.72410059 115.92496  -88.51002335 158.71089
## 2005.6822       35.30470 -45.58591678 116.19531  -88.40682134 159.01622
## 2005.6849       35.36310 -45.59354682 116.31974  -88.44940462 159.17560
## 2005.6877       33.27493 -47.74768237 114.29755  -90.63846494 157.18833
## 2005.6904       33.36470 -47.72384128 114.45323  -90.64952020 157.37891
## 2005.6932       32.42228 -48.73212016 113.57669  -91.69266708 156.53724
## 2005.6959       32.80239 -48.41783190 114.02261  -91.41321855 157.01799
## 2005.6986       31.29048 -49.99550212 112.57646  -93.02570029 155.60665
## 2005.7014       32.10386 -49.24782526 113.45555  -92.31280681 156.52053
## 2005.7041       32.82214 -48.59519618 114.23949  -91.69493303 157.33922
## 2005.7068       32.79602 -48.68692208 114.27896  -91.82138623 157.41343
## 2005.7096       32.64115 -48.90733975 114.18964  -92.07650327 157.35881
## 2005.7123       31.83624 -49.77774333 113.45023  -92.98157835 156.65407
## 2005.7151       32.26565 -49.41378516 113.94508  -92.65226387 157.18355
## 2005.7178       33.52658 -48.21823950 115.27140  -91.49133417 158.54450
## 2005.7205       33.56537 -48.24479072 115.37553  -91.55247368 158.68321
## 2005.7233       33.47551 -48.39993304 115.35096  -91.74217670 158.69321
## 2005.7260       33.41300 -48.52767857 115.35369  -91.90445538 158.73046
## 2005.7288       30.62072 -51.38514841 112.62658  -94.79643090 156.03786
## 2005.7315       30.19323 -51.87776127 112.26423  -95.32352204 155.70999
## 2005.7342       30.18195 -51.95412949 112.31802  -95.43434119 155.79823
## 2005.7370       29.55750 -52.64360345 111.75860  -96.15823881 155.27324
## 2005.7397       31.34930 -50.91678116 113.61538  -94.46581298 157.16441
## 2005.7425       32.07855 -50.25245435 114.40955  -93.83585547 157.99296
## 2005.7452       33.08041 -49.31546403 115.47629  -92.93320738 159.09404
## 2005.7479       32.90438 -49.55632516 115.36508  -93.20838371 159.01714
## 2005.7507       32.97420 -49.55127529 115.49967  -93.23762209 159.18602
## 2005.7534       32.59299 -49.99720419 115.18319  -93.71781235 158.90379
## 2005.7562       30.89193 -51.76293696 113.54679  -95.51777965 157.30164
## 2005.7589       29.06668 -53.65280544 111.78617  -97.44185590 155.57522
## 2005.7616       27.82460 -54.95944997 110.60866  -98.78268149 154.43189
## 2005.7644       29.49235 -53.35622535 112.34092  -97.21361130 156.19831
## 2005.7671       30.61111 -52.30193385 113.52415  -96.19344765 157.41567
## 2005.7699       31.43556 -51.54190031 114.41302  -95.46751544 158.33864
## 2005.7726       32.41830 -50.62353424 115.46013  -94.58322425 159.41982
## 2005.7753       32.49348 -50.61266554 115.59964  -94.60640404 159.59337
## 2005.7781       32.00232 -51.16809842 115.17274  -95.19585907 159.20050
## 2005.7808       30.64694 -52.58769888 113.88158  -96.64945542 157.94334
## 2005.7836       30.68177 -52.61704325 113.98058  -96.71276947 158.07630
## 2005.7863       30.74598 -52.61694814 114.10891  -96.74661789 158.23858
## 2005.7890       28.20937 -55.21762928 111.63637  -99.38121646 155.79996
## 2005.7918       27.97080 -55.52022593 111.46182  -99.71770453 155.65930
## 2005.7945       28.74415 -54.81084721 112.29915  -99.04219125 156.53049
## 2005.7973       28.98285 -54.63607480 112.60177  -98.90125837 156.86695
## 2005.8000       30.53876 -53.14404152 114.22155  -97.44303878 158.52055
## 2005.8027       31.08754 -52.65908350 114.83417  -96.99186864 159.16695
## 2005.8055       30.40679 -53.40361624 114.21719  -97.77016355 158.58374
## 2005.8082       29.89525 -53.97887945 113.76939  -98.37916324 158.16967
## 2005.8110       29.66012 -54.27769027 113.59794  -98.71168494 158.03193
## 2005.8137       30.53320 -53.46824479 114.53465  -97.93592478 159.00233
## 2005.8164       29.17009 -54.89493914 113.23512  -99.39627895 157.73646
## 2005.8192       28.24376 -55.88480459 112.37233 -100.41977878 156.90731
## 2005.8219       27.29625 -56.89580994 111.48831 -101.46439312 156.05689
## 2005.8247       28.10239 -56.15311146 112.35789 -100.75527832 156.96005
## 2005.8274       28.14285 -56.17604087 112.46174 -100.81176613 157.09747
## 2005.8301       29.10980 -55.27243447 113.49204  -99.94169292 158.16130
## 2005.8329       26.81817 -57.62736544 111.26371 -102.33013193 155.96647
## 2005.8356       25.08926 -59.41952652 109.59805 -104.15577595 154.33430
## 2005.8384       27.16000 -57.41199326 111.73199 -102.18170058 156.50169
## 2005.8411       28.63714 -55.99800458 113.27229 -100.80114482 158.07543
## 2005.8438       27.33552 -57.36273676 112.03377 -102.19928498 156.87032
## 2005.8466       24.49701 -60.26430443 109.25833 -105.13423576 154.12826
## 2005.8493       25.26265 -59.56167969 110.08699 -104.46496931 154.99028
## 2005.8521       25.11165 -59.77564684 109.99896 -104.71226998 154.93558
## 2005.8548       25.35216 -59.59806382 110.30238 -104.56799578 155.27232
## 2005.8575       26.19308 -58.82001560 111.20618 -103.82323173 156.20940
## 2005.8603       29.20590 -55.87002631 114.28183 -100.90650201 159.31830
## 2005.8630       26.40150 -58.73720749 111.54021 -103.80691821 156.60992
## 2005.8658       24.21487 -60.98657361 109.41632 -106.08949486 154.51924
## 2005.8685       25.53978 -59.72435567 110.80392 -104.86046303 155.94002
## 2005.8712       26.85189 -58.47488905 112.17867 -103.64415811 157.34794
## 2005.8740       24.90127 -60.48811124 110.29065 -105.69051769 155.49305
## 2005.8767       23.90887 -61.54306267 109.36080 -106.77858224 154.59632
## 2005.8795       23.80878 -61.70566087 109.32321 -106.97426932 154.59182
## 2005.8822       25.05204 -60.52486229 110.62893 -105.82653547 155.93061
## 2005.8849       23.43831 -62.20099985 109.07763 -107.53571363 154.41234
## 2005.8877       23.76433 -61.93735296 109.46601 -107.30508329 154.83374
## 2005.8904       25.34406 -60.41995116 111.10806 -105.82067402 156.50879
## 2005.8932       26.28641 -59.53987656 112.11270 -104.97356800 157.54639
## 2005.8959       24.99060 -60.89791863 110.87912 -106.36455474 156.34576
## 2005.8986       25.35943 -60.59127440 111.31014 -106.09083132 156.80970
## 2005.9014       26.16791 -59.84494222 112.18076 -105.37739615 157.71322
## 2005.9041       24.76349 -61.31146062 110.83844 -106.87678781 156.40377
## 2005.9068       25.16508 -60.97192259 111.30209 -106.57009935 156.90027
## 2005.9096       27.95376 -58.24525420 114.15278 -103.87625687 159.78378
## 2005.9123       25.43331 -60.82767496 111.69429 -106.49147994 157.35809
## 2005.9151       23.22653 -63.09636761 109.54944 -108.79295137 155.24602
## 2005.9178       23.51102 -62.87375722 109.89580 -108.60309625 155.62514
## 2005.9205       22.96464 -63.48196476 109.41125 -109.24403561 155.17333
## 2005.9233       24.85602 -61.65237292 111.36442 -107.44715221 157.15920
## 2005.9260       23.18303 -63.38710598 109.75318 -109.21457036 155.58064
## 2005.9288       23.64472 -62.98712193 110.27656 -108.84724811 156.13669
## 2005.9315       20.78873 -65.90477063 107.48222 -111.79753535 153.37499
## 2005.9342       22.33401 -64.42110261 109.08911 -110.34648269 155.01449
## 2005.9370       22.45518 -64.36149297 109.27186 -110.31946525 155.22983
## 2005.9397       23.90995 -62.96825089 110.78815 -108.95879228 156.77869
## 2005.9425       23.64067 -63.29901110 110.58035 -109.32209855 156.60344
## 2005.9452       21.31118 -65.68994214 108.31230 -111.74555266 154.36791
## 2005.9479       21.61086 -65.45165452 108.67337 -111.53976514 154.76148
## 2005.9507       21.30156 -65.82230549 108.42542 -111.94289332 154.54601
## 2005.9534       20.28882 -66.89635360 107.47399 -113.04939579 153.62703
## 2005.9562       20.95629 -66.29015069 108.20272 -112.47562442 154.38820
## 2005.9589       21.13757 -66.17009127 108.44523 -112.38797380 154.66311
## 2005.9616       20.04842 -67.32041923 107.41726 -113.57068784 153.66752
## 2005.9644       18.73040 -68.69956924 106.16038 -114.98220127 152.44301
## 2005.9671       16.53958 -70.95149103 104.03064 -117.26646387 150.34561
## 2005.9699       17.33594 -70.21617233 104.88806 -116.56346341 151.23535
## 2005.9726       17.47624 -70.13688787 105.08936 -116.51647466 151.46895
## 2005.9753       20.61486 -67.05923453 108.28895 -113.47109457 154.70081
## 2005.9781       21.01791 -66.71710842 108.75292 -113.16121928 155.19703
## 2005.9808       21.36981 -66.42608434 109.16571 -112.90242364 155.64204
## 2005.9836       22.84619 -65.01054665 110.70292 -111.51909206 157.21147
## 2005.9863       20.07386 -67.84366814 107.99139 -114.38439737 154.53212
## 2005.9890       21.32119 -66.65709859 109.29947 -113.22998940 155.87236
## 2005.9918       21.08895 -66.95004654 109.12795 -113.55507674 155.73298
## 2005.9945       22.44464 -65.65502767 110.54431 -112.29217511 157.18146
## 2005.9973       23.88663 -64.27366883 112.04693 -110.94291140 158.71617
## 2006.0000       24.41530 -63.80558256 112.63619 -110.50689821 159.33750
## 2006.0027       23.64690 -64.63452859 111.92833 -111.36789530 158.66170
## 2006.0055       22.56934 -65.77260055 110.91127 -112.53799637 157.67667

NOTE: AIC = 21549.73 , BIC = 21566.81 , RMSE = 2.902054

Also, the headline of the plot below contains STL + ETS(A,N,N) showing that the automated model characterized error of the “tsT_PK” SERIES as Additative (A), trend as None (A) and seasonality as None (N) which is equivalent to simple exponential model(ses) with only alpha parameter i.e level.

#Model 2 : ARIMA Model It is used for forecasting for both seasonal and non- seasonal level.

#Changing to stationary set

train.sta1 <- diff(train1, diff =2)
test.sta1 <- diff(test1, diff =2)

2.1 Auto Arima model without sesonality:

set.seed(301)
model_VAR2 <- auto.arima(train.sta1, seasonal = FALSE)
summary(model_VAR2)
## Series: train.sta1 
## ARIMA(1,0,3) with non-zero mean 
## 
## Coefficients:
##          ar1      ma1     ma2     ma3   mean
##       0.4847  -1.7074  0.5080  0.2024  0e+00
## s.e.  0.0331   0.0342  0.0617  0.0285  4e-04
## 
## sigma^2 estimated as 8.919:  log likelihood=-5507.16
## AIC=11026.31   AICc=11026.35   BIC=11060.46
## 
## Training set error measures:
##                        ME     RMSE      MAE MPE MAPE      MASE       ACF1
## Training set 0.0004542817 2.983101 2.070462 NaN  Inf 0.4204283 0.01071112

NOTE: AIC = 11026.31 , BIC = 11060.46 , RMSE =2.983101

tsdisplay(residuals(model_VAR2), lag.max = 45, main = '(1,1,1) Model Residuals')

#2.2 Auto Arima model with sesonality:

set.seed(301)
model2_VAR2 <- auto.arima(train.sta1, seasonal= TRUE)
summary(model2_VAR2)
## Series: train.sta1 
## ARIMA(5,0,0) with zero mean 
## 
## Coefficients:
##           ar1      ar2      ar3      ar4      ar5
##       -0.8365  -0.7683  -0.6262  -0.4388  -0.1951
## s.e.   0.0210   0.0259   0.0276   0.0259   0.0210
## 
## sigma^2 estimated as 12.74:  log likelihood=-5894.48
## AIC=11800.97   AICc=11801   BIC=11835.12
## 
## Training set error measures:
##                        ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set 0.0005491905 3.564718 2.401756 NaN  Inf 0.4877009 -0.04422588

NOTE: AIC = 11800.97 , BIC = 11835.12 , RMSE =3.564718

tsdisplay(residuals(model2_VAR2), lag.max = 20, main = 'Seasonal Model Residuals')

NOTE: Above Graphs shows serious lags at 5, so modify model for p or q = 5

#2.3 Manual Arima

Passing parameter order = (1,0,5) which means p(member of lags of a variable to be used as a predictor) = 1, d = 0(members of differences needed for stationarity) and q = 5 (moving average order - number of lagged forecasts error in the prediction equation)

set.seed(301)
model1_VAR2 <- arima(train.sta1, order = c(1,0,5))
summary(model1_VAR2)
## 
## Call:
## arima(x = train.sta1, order = c(1, 0, 5))
## 
## Coefficients:
##           ar1      ma1      ma2     ma3     ma4     ma5  intercept
##       -0.0608  -1.1492  -0.1639  0.1007  0.1046  0.1133      0e+00
## s.e.   0.2824   0.2810   0.3490  0.0430  0.0517  0.0600      3e-04
## 
## sigma^2 estimated as 8.885:  log likelihood = -5505.47,  aic = 11026.94
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE          ACF1
## Training set 0.002088656 2.980804 2.069952 NaN  Inf 0.3813413 -0.0001374768

NOTE: AIC = 11026.94, RMSE =2.980804

tsdisplay(residuals(model1_VAR2), lag.max = 20, main = 'Seasonal Model Residuals')

#Visualising forecast by each model

par(mfrow = c(2,2))
fit_auto_train_VAR2 %>% forecast(h=341) %>% autoplot() + autolayer(test1) 

model1_VAR2 %>% forecast(h=341) %>% autoplot() 

model2_VAR2 %>% forecast(h=341) %>% autoplot() 

model_VAR2 %>% forecast(h=341) %>% autoplot() 

Plot and comment on the residuals of the fitted data for both models in the same plot.

Residual Plot - to confirm no problem with this model

par(mfrow= c(2,2))
hist(fit_auto_train_VAR2$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(fit_auto_train_VAR2$residuals))

hist(model_VAR2$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model_VAR2$residuals))

hist(model1_VAR2$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model1_VAR2$residuals))

hist(model2$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model2_VAR2$residuals))

OBSERVATION: The residuals for each model are normally distributed, which shows the good fit of the models. It is normally distrbuted.

Testing the autocorrelation and partial autocorrelation of the residuals (using the plots and Ljung-Box Q statistic) up to an appropriate lag *

  1. ACF & PACF Plots
par(mfrow = c(2,2))
acf(fit_auto_train_VAR2$residuals, main = 'Correlogram')
acf(model_VAR2$residuals, main = 'Correlogram')
acf(model1_VAR2$residuals, main = 'Correlogram')
acf(model2_VAR2$residuals, main = 'Correlogram')

#PACF PLOTS

par(mfrow = c(2,2))
pacf(fit_auto_train_VAR2$residuals, main = 'Partial Corelogram')
pacf(model_VAR2$residuals, main = 'Partial Corelogram')
pacf(model1_VAR2$residuals, main = 'Partial Corelogram')
pacf(model2_VAR2$residuals, main = 'Partial Corelogram')

NOTE: 1. Residuals are the difference between actaual and fitted values
2. These blue lines are signficance bounds
3. This graph shows us the autocorelations for insample forecast errors
4. Do not exceed these significance bounds for lags 1 to end

  1. Ljunx-Box Test to check the autocorelations of the residuals

This test used to test the lack of fit of a time series model.
This test is applied to the residuals of the time series model

We reject the null hypothesis, if p value is less than 0.05. This test is generally done for ARIMA models

#Ljunx-Box Test to check the autocorelations of the residuals

Box.test(model_VAR2$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model_VAR2$residuals
## X-squared = 46.055, df = 20, p-value = 0.0007922
Box.test(model1_VAR2$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model1_VAR2$residuals
## X-squared = 39.927, df = 20, p-value = 0.005103
Box.test(model2_VAR2$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model2_VAR2$residuals
## X-squared = 261.08, df = 20, p-value < 2.2e-16

OBSERVATIONS:
For above performed Ljunx-Box Test on model_VAR2,model1_VAR2 and model2_VAR2 output, we can say that p-value is less than significance level 0.05, then there is a statistical significance.
We conclude that there is a liitle evidence of non-zero auto-corelations in the insample forecast errors at lags 1 to 20.
So we rejects the null hypothesis. Hence, we conclude that there is serial auto corelation present between lags.

Validating The Model by testing the model performance with Holdout set

Here, choosing RMSE as evaluation parameter. The lower RMSE indicates a more accurate forecast.

Below, is the comparison of model. We are fitting the model to test set which is the holdout set. And checking the accuracy using RMSE(Root mean square error) evaluation parameter.

accuracy(forecast(fit_auto_train_VAR2), test1) ["Test set", "RMSE"]
## [1] 6.771375
#o/p - 6.771375
accuracy(forecast(model1_VAR2), test.sta1) ["Test set", "RMSE"]
## [1] 4.442126
#O/P - 4.442126
accuracy(forecast(model2_VAR2), test.sta1) ["Test set", "RMSE"]
## [1] 4.453695
#o/p - 4.453695
accuracy(forecast(model_VAR2), test.sta1) ["Test set", "RMSE"]
## [1] 4.44153
#O/P - 4.44153

NOTE:- Here, the RMSE value is less of ARIMA models compared to exponential model.
The minimum RMSE is of model1_VAR2.
The RMSE of auto arima models with seasonality and without seasonality have little difference.

COMMENT ON BEST MODEL AND ITS PARAMETERS We observed the best model to be ARIMA model i.e.“model1_VAR2”.

summary(model1_VAR2)
## 
## Call:
## arima(x = train.sta1, order = c(1, 0, 5))
## 
## Coefficients:
##           ar1      ma1      ma2     ma3     ma4     ma5  intercept
##       -0.0608  -1.1492  -0.1639  0.1007  0.1046  0.1133      0e+00
## s.e.   0.2824   0.2810   0.3490  0.0430  0.0517  0.0600      3e-04
## 
## sigma^2 estimated as 8.885:  log likelihood = -5505.47,  aic = 11026.94
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE          ACF1
## Training set 0.002088656 2.980804 2.069952 NaN  Inf 0.3813413 -0.0001374768

Passing parameter order = (1,0,5) ,which means

p = 1,
d = 0
q = 5

For the best model, 1 member of lag is used as a predictor. As we transformed the series to stationary so members of differences needed for stationarity is 0. 5 are the number of lagged forecasts errors used in the prediction equation.

AIC = 11026.94, RMSE = 2.980804 which is minimum as compared to other models

3. Forecasting on Average T (T_AV)

#Checking the attributes of time series

attributes(tsT_AV)
## $tsp
## [1] 1998.00 2004.94  365.00
## 
## $class
## [1] "ts"

Plotting the series

plot(tsT_AV, main = "Average T observed for years (tsT_AV)", ylim = c(0,35), col = "red")

From the plot, we observes that there is no trend but seems to have seasonlity.
Most of the data is missing for year 2002 to 2003 .

Checking the statistics and visualising the missing data in the series:

plotNA.distribution(tsT_AV)

As we see there is a large gap in the missing data, so locf and NOCB would not seems the good technique to implement here.

Also, as no trend is being observed here, so not necessary to use “Kalman filter with arima state space model” as it works best for series having trend and seasonality.

Hence, using “Linear interpolation” technique to impute missing values. This method works good for series having seasonality but no trend type of time series data.

#Impute the missing values with “na_interpolation” function of “imputeTS package” and visualise the imputed values in the time series

tsT_AV.imp <- na_interpolation(tsT_AV, method = "linear")
plotNA.imputations(tsT_AV, tsT_AV.imp)

#Decomposition Of Time Series

decomp2 <- decompose(tsT_AV.imp)
plot(decomp2, col = "red")

OBSERVATIONS:
1)There seems to have trend and seasonality both
2)From 1998 to 2001 seems to have downward trend and from 2001 to 2005 seems to have upward trend.
3)We observe some seasonality in the series. Means there is some seasonality. Seasonality describes cyclical effects due to the time of the year.

Let’s explore this seasonality pattern by plotting the decomp$seasonal components such that y-axis represents Seasonality Index and x-axis has month

#Visualising the seasonal data

par(mfrow= c(1,2))
plot(decomp2$seasonal, type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")
plot(decomp2$seasonal[1:400], type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")

OBSERVATIONS:
1) There is increase in the mid of months of a year.
2) There is a drop in peak T for starting and ending months of the year

#Stationarity check of time series
We could check the stationarity of time series by ADF and KPSS Tests:

  1. Augmented Dickey- Fuller(ADF) t-statistic test for unit root
  2. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) for level or trend stationarity
adf.test(tsT_AV.imp)
## 
##  Augmented Dickey-Fuller Test
## 
## data:  tsT_AV.imp
## Dickey-Fuller = -3.7095, Lag order = 13, p-value = 0.02351
## alternative hypothesis: stationary
#p-value is less than 0.05, hence it is stationary
kpss.test(tsT_AV.imp, null="Trend")
## 
##  KPSS Test for Trend Stationarity
## 
## data:  tsT_AV.imp
## KPSS Trend = 0.19928, Truncation lag parameter = 8, p-value = 0.01627
#p-value is less than 0.05, hence it is not trend- stationary

NOTE: From the both tests results, it leads to CASE4, we could recheck by visualising it i.e. by plotting ACF. Second, we can check each for characteristics of stationarity by looking at the autocorrelation functions (ACF) of each signal. For a stationary signal, because we expect no dependence with time, we would expect the ACF to go to 0 for each time lag (τ). Lets visualize the signals and ACFs.

acf(tsT_AV.imp,lag.max = length(tsT_AV.imp),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

It seems difference of 1 is sufficient to make the series stationary.

tsT_AV.sta <- diff(tsT_AV.imp, diff =1)
acf(tsT_AV.sta, lag.max = length(tsT_AV.sta),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Hence, after differencing our series is now transformed to stationary time series.

#Visualising the stationary series

plot(tsT_AV.sta, col = "red", main = "Stationary Series")

#CREATING MODELS AND FORECASTING: USING EXPONENTIAL SMOOTHING AND ARIMA TECHNIQUES FOR TIME SERIES FORECASTING

1.Train - Test Split of the data

train2 <- window(tsT_AV.imp, end = c(2004, 3))
test2 <- window(tsT_AV.imp, start = c(2004, 4))

#Model 1 : Auto Exponential Smoothening Model

set.seed(301)
fit_auto_train_Var3 <- forecast(train2)
summary(fit_auto_train_Var3)
## 
## Forecast method: STL +  ETS(A,N,N)
## 
## Model Information:
## ETS(A,N,N) 
## 
## Call:
##  ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend) 
## 
##   Smoothing parameters:
##     alpha = 0.9999 
## 
##   Initial states:
##     l = 21.3691 
## 
##   sigma:  2.557
## 
##      AIC     AICc      BIC 
## 20992.54 20992.55 21009.62 
## 
## Error measures:
##                       ME    RMSE      MAE       MPE     MAPE     MASE      ACF1
## Training set 0.002453102 2.55584 1.759238 -3.015496 13.56144 0.421226 0.0730533
## 
## Forecasts:
##           Point Forecast        Lo 80     Hi 80        Lo 95     Hi 95
## 2004.0082       18.06044  14.78350513  21.33738   13.0488004  23.07208
## 2004.0110       18.87994  14.24588158  23.51399   11.7927614  25.96711
## 2004.0137       18.71912  13.04368482  24.39456   10.0392885  27.39896
## 2004.0164       20.65887  14.10549253  27.21225   10.6363434  30.68140
## 2004.0192       17.99232  10.66545156  25.31918    6.7868443  29.19779
## 2004.0219       16.02033   7.99418310  24.04648    3.7453959  28.29527
## 2004.0247       20.11563  11.44641754  28.78484    6.8572138  33.37405
## 2004.0274       21.98155  12.71378748  31.24931    7.8077311  36.15537
## 2004.0301       21.76538  11.93545311  31.59532    6.7318017  36.79897
## 2004.0329       19.58277   9.22112606  29.94442    3.7360020  35.42954
## 2004.0356       18.75602   7.88863947  29.62339    2.1357980  35.37623
## 2004.0384       19.47598   8.12538457  30.82658    2.1167422  36.83522
## 2004.0411       20.18772   8.37365074  32.00178    2.1196615  38.25577
## 2004.0438       18.85494   6.59490960  31.11497    0.1048418  37.60504
## 2004.0466       19.79636   7.10602989  32.48669    0.3881747  39.20455
## 2004.0493       18.03219   4.92568342  31.13871   -2.0124847  38.07687
## 2004.0521       16.92514   3.41526164  30.43502   -3.7364357  37.58671
## 2004.0548       18.45992   4.55837292  32.36146   -2.8006605  39.72050
## 2004.0575       20.17406   5.89158206  34.45653   -1.6691039  42.01722
## 2004.0603       19.23714   4.58363750  33.89065   -3.1734605  41.64775
## 2004.0630       19.73808   4.72270773  34.75345   -3.2259506  42.70211
## 2004.0658       19.50550   4.13678384  34.87423   -3.9989255  43.00993
## 2004.0685       17.90914   2.19501780  33.62327   -6.1235377  41.94182
## 2004.0712       19.15798   3.10588300  35.21008   -5.3915851  43.70755
## 2004.0740       21.87553   5.49242964  38.25863   -3.1802609  46.93133
## 2004.0767       20.53740   3.82985132  37.24495   -5.0145910  46.08939
## 2004.0795       20.19807   3.17225357  37.22388   -5.8406682  46.23680
## 2004.0822       22.20058   4.86234514  39.53882   -4.3159639  48.71713
## 2004.0849       23.09218   5.44704563  40.73731   -3.8937228  50.07808
## 2004.0877       23.81598   5.86920433  41.76276   -3.6312459  51.26321
## 2004.0904       23.59367   5.35023262  41.83711   -4.3072594  51.49460
## 2004.0932       20.37457   1.83922321  38.90992   -7.9727975  48.72194
## 2004.0959       18.87838   0.05564178  37.70111   -9.9085114  47.66526
## 2004.0986       18.53407  -0.57172769  37.63986  -10.6857253  47.75386
## 2004.1014       19.23206  -0.15266143  38.61679  -10.4143156  48.87845
## 2004.1041       20.80432   1.14462373  40.46402   -9.2625923  50.87124
## 2004.1068       22.68452   2.75364049  42.61540   -7.7971293  53.16617
## 2004.1096       21.17052   0.97210517  41.36894   -9.7202913  52.06134
## 2004.1123       21.41336   0.95089793  41.87582   -9.8812736  52.70799
## 2004.1151       23.78461   3.06146980  44.50774   -7.9086960  55.47791
## 2004.1178       22.95430   1.97372046  43.93487   -9.1327253  55.04132
## 2004.1205       19.52718  -1.70771447  40.76207  -12.9487880  52.00315
## 2004.1233       20.44717  -1.03902995  41.93337  -12.4131380  53.30748
## 2004.1260       21.06967  -0.66493174  42.80428  -12.1705361  54.30988
## 2004.1288       20.29387  -1.68632462  42.27407  -13.3219393  53.90969
## 2004.1315       20.69749  -1.52559457  42.92057  -13.2897830  54.68476
## 2004.1342       19.67898  -2.78435740  42.14232  -14.6757294  54.03369
## 2004.1370       19.71554  -2.98551170  42.41659  -15.0027213  54.43380
## 2004.1397       21.06769  -1.86860871  44.00399  -14.0103517  56.14573
## 2004.1425       23.22396   0.05480506  46.39312  -12.2102071  58.65814
## 2004.1452       23.90914   0.50943289  47.30884  -11.8776217  59.69589
## 2004.1479       25.24340   1.61540284  48.87140  -10.8925034  61.37930
## 2004.1507       24.71145   0.85734647  48.56556  -11.7702549  61.19316
## 2004.1534       22.57550  -1.50259109  46.65360  -14.2487637  59.39977
## 2004.1562       22.35381  -1.94620413  46.65383  -14.8098550  59.51748
## 2004.1589       22.88516  -1.63476971  47.40509  -14.6148357  60.38515
## 2004.1616       22.18454  -2.55334340  46.92243  -15.6487896  60.01788
## 2004.1644       22.40008  -2.55386400  47.35402  -15.7636827  60.56384
## 2004.1671       25.00777  -0.16037294  50.17591  -13.4835823  63.49912
## 2004.1699       24.79341  -0.58712776  50.17394  -14.0227709  63.60958
## 2004.1726       24.51304  -1.07812323  50.10420  -14.6252670  63.65135
## 2004.1753       24.73004  -1.07003504  50.53011  -14.7277692  64.18785
## 2004.1781       23.01030  -2.99700294  49.01761  -16.7644392  62.78505
## 2004.1808       19.34047  -6.87243035  45.55337  -20.7487015  59.42964
## 2004.1836       19.77775  -6.63914646  46.19464  -20.6234054  60.17890
## 2004.1863       20.44623  -6.17309130  47.06556  -20.2645106  61.15698
## 2004.1890       22.58076  -4.23946736  49.40099  -18.4372381  63.59876
## 2004.1918       25.06703  -1.95260294  52.08667  -16.2559345  66.39000
## 2004.1945       24.72328  -2.49430822  51.94086  -16.9024272  66.34898
## 2004.1973       23.85013  -3.56397314  51.26423  -18.0761229  65.77638
## 2004.2000       23.35898  -4.25024774  50.96820  -18.8656878  65.58364
## 2004.2027       24.59989  -3.20308239  52.40287  -17.9210879  67.12087
## 2004.2055       26.31226  -1.68312878  54.30764  -16.5029900  69.12750
## 2004.2082       24.93436  -3.25211865  53.12084  -18.1731402  68.04186
## 2004.2110       23.10982  -5.26647174  51.48611  -20.2879724  66.50761
## 2004.2137       23.14327  -5.42156892  51.70811  -20.5428811  66.82942
## 2004.2164       24.91530  -3.83685172  53.66745  -19.0573208  68.88792
## 2004.2192       24.64446  -4.29378880  53.58271  -19.6127730  68.90170
## 2004.2219       24.37121  -4.75195240  53.49437  -20.1688222  68.91124
## 2004.2247       25.42344  -3.88346491  54.73034  -19.3976028  70.24448
## 2004.2274       25.97598  -3.51352130  55.46548  -19.1243212  71.07628
## 2004.2301       27.92252  -1.74845620  57.59350  -17.4553232  73.30037
## 2004.2329       27.16013  -2.69122227  57.01148  -18.4935724  72.81383
## 2004.2356       27.41139  -2.61925183  57.44203  -18.5165116  73.33929
## 2004.2384       28.75407  -1.45479698  58.96293  -17.4464031  74.95454
## 2004.2411       28.01375  -2.37228816  58.39980  -18.4576873  74.48520
## 2004.2438       27.64453  -2.91766587  58.20672  -19.0963142  74.38537
## 2004.2466       26.55936  -4.17797805  57.29670  -20.4493413  73.56806
## 2004.2493       28.07138  -2.84010822  58.98287  -19.2036610  75.34642
## 2004.2521       29.88792  -1.19673925  60.97258  -17.6519651  77.42781
## 2004.2548       29.35958  -1.89730001  60.61645  -18.4436911  77.16284
## 2004.2575       27.81721  -3.61094164  59.24535  -20.2479983  75.88241
## 2004.2603       27.63597  -3.96252545  59.23446  -20.6897564  75.96169
## 2004.2630       29.13788  -2.63003675  60.90581  -19.4469583  77.72273
## 2004.2658       29.67549  -2.26096474  61.61194  -19.1671012  78.51808
## 2004.2685       29.39689  -2.70720578  61.50099  -19.7020888  78.49587
## 2004.2712       28.35751  -3.91336623  60.62838  -20.9965348  77.71155
## 2004.2740       29.00005  -3.43674147  61.43684  -20.6077416  78.60784
## 2004.2767       27.59868  -5.00318529  60.20054  -22.2615700  77.45893
## 2004.2795       27.82218  -4.94392219  60.58829  -22.2892513  77.93362
## 2004.2822       29.22809  -3.70143756  62.15762  -21.1332774  79.58946
## 2004.2849       30.13013  -2.96201400  63.22228  -20.4799374  80.74020
## 2004.2877       30.72087  -2.53309942  63.97483  -20.1366854  81.57842
## 2004.2904       30.04487  -3.37013474  63.45987  -21.0589684  81.14870
## 2004.2932       30.32132  -3.25394537  63.89659  -21.0276179  81.67026
## 2004.2959       30.57247  -3.16230125  64.30724  -21.0204096  82.16535
## 2004.2986       30.29348  -3.60003826  64.18701  -21.5421850  82.12915
## 2004.3014       29.67076  -4.38077309  63.72230  -22.4065665  81.74809
## 2004.3041       29.88011  -4.32870630  64.08893  -22.4377601  82.19798
## 2004.3068       31.41306  -2.95232192  65.77844  -21.1442549  83.97037
## 2004.3096       31.96973  -2.55150346  66.49096  -20.8259398  84.76540
## 2004.3123       31.43152  -3.24486270  66.10791  -21.6014316  84.46447
## 2004.3151       31.07754  -3.75330835  65.90838  -22.1916440  84.34672
## 2004.3178       30.93141  -4.05321003  65.91604  -22.5729514  84.43578
## 2004.3205       31.02438  -4.11335486  66.16211  -22.7141457  84.76290
## 2004.3233       31.56732  -3.72284781  66.85750  -22.4043364  85.53898
## 2004.3260       31.30766  -4.13429777  66.74962  -22.8961371  85.51146
## 2004.3288       29.25900  -6.33409444  64.85210  -25.1759418  83.69395
## 2004.3315       29.20720  -6.53639461  64.95080  -25.4579118  83.87232
## 2004.3342       30.20564  -5.68783009  66.09910  -24.6886830  85.09995
## 2004.3370       32.03071  -4.01199693  68.07343  -23.0918557  87.15328
## 2004.3397       30.49587  -5.69547118  66.68721  -24.8540099  85.84575
## 2004.3425       30.30228  -6.03708256  66.64164  -25.2739795  85.87854
## 2004.3452       29.46330  -7.02348810  65.95008  -26.3384254  85.26502
## 2004.3479       30.11734  -6.51627229  66.75095  -25.9089359  86.14362
## 2004.3507       31.44472  -5.33513969  68.22457  -24.8052193  87.69465
## 2004.3534       30.93522  -5.99030403  67.86073  -25.5374931  87.40792
## 2004.3562       31.74299  -5.32762224  68.81360  -24.9516177  88.43759
## 2004.3589       32.45148  -4.76365521  69.66661  -24.4641576  89.36712
## 2004.3616       32.15963  -5.19946973  69.51873  -24.9761832  89.29545
## 2004.3644       31.91213  -5.59038808  69.41464  -25.4430200  89.26727
## 2004.3671       32.14132  -5.50406145  69.78670  -25.4323226  89.71496
## 2004.3699       32.35224  -5.43546504  70.13995  -25.4390695  90.14356
## 2004.3726       32.18983  -5.73966716  70.11933  -25.8183322  90.19800
## 2004.3753       32.69823  -5.37253481  70.76900  -25.5259808  90.92244
## 2004.3781       32.14524  -6.06626436  70.35675  -26.2942149  90.58470
## 2004.3808       33.10437  -5.24736595  71.45610  -25.5495477  91.75828
## 2004.3836       33.69082  -4.80062765  72.18227  -25.1767701  92.55841
## 2004.3863       33.61506  -5.01559678  72.24572  -25.4654325  92.69556
## 2004.3890       32.89056  -5.87881214  71.65993  -26.4020765  92.18319
## 2004.3918       31.33293  -7.57465080  70.24052  -28.1710820  90.83695
## 2004.3945       31.81424  -7.23107430  70.85955  -27.9004133  91.52889
## 2004.3973       31.79357  -7.38897831  70.97613  -28.1309689  91.71812
## 2004.4000       32.67250  -6.64681294  71.99182  -27.4612015  92.80621
## 2004.4027       32.57865  -6.87695239  72.03426  -27.7634880  92.92079
## 2004.4055       33.75419  -5.83723784  73.34561  -26.7956722  94.30405
## 2004.4082       34.36927  -5.35750639  74.09605  -26.3875936  95.12614
## 2004.4110       34.47923  -5.38244464  74.34091  -26.4839414  95.44240
## 2004.4137       34.61665  -5.37946379  74.61277  -26.5521293  95.78544
## 2004.4164       34.80720  -5.32290644  74.93731  -26.5665022  96.18090
## 2004.4192       33.55523  -6.70842054  73.81888  -28.0227106  95.13317
## 2004.4219       33.99493  -6.40182831  74.39168  -27.7865789  95.77643
## 2004.4247       32.81941  -7.71001367  73.34883  -29.1649935  94.80381
## 2004.4274       32.51460  -8.14705303  73.17626  -29.6720329  94.70124
## 2004.4301       32.94550  -7.84795495  73.73896  -29.4427079  95.33372
## 2004.4329       33.28041  -7.64442993  74.20525  -29.3087313  95.86955
## 2004.4356       33.92599  -7.12980744  74.98179  -28.8634346  96.71542
## 2004.4384       33.27267  -7.91367296  74.45901  -29.7164056  96.26174
## 2004.4411       33.49552  -7.82094826  74.81200  -29.6925680  96.68362
## 2004.4438       33.17568  -8.27051588  74.62187  -30.2108064  96.56217
## 2004.4466       32.94135  -8.63416582  74.51686  -30.6429128  96.52561
## 2004.4493       33.06158  -8.64284539  74.76601  -30.7198367  96.84300
## 2004.4521       33.92229  -7.91065445  75.75524  -30.0556797  97.90027
## 2004.4548       32.91469  -9.04637986  74.87577  -31.2592307  97.08862
## 2004.4575       32.56952  -9.51928668  74.65833  -31.7997566  96.93880
## 2004.4603       33.65607  -8.56009244  75.87222  -30.9079768  98.22011
## 2004.4630       33.67842  -8.66470004  76.02155  -31.0797962  98.43665
## 2004.4658       33.45668  -9.01303313  75.92639  -31.4951401  98.40850
## 2004.4685       34.09710  -8.49882616  76.69302  -31.0477448  99.24194
## 2004.4712       34.21924  -8.50251875  76.94100  -31.1180517  99.55653
## 2004.4740       34.35679  -8.49043921  77.20401  -31.1723908  99.88597
## 2004.4767       34.02227  -8.95005589  76.99460  -31.6982322  99.74278
## 2004.4795       34.53089  -8.56617492  77.62796  -31.3803838 100.44217
## 2004.4822       34.78040  -8.44104898  78.00184  -31.3210998 100.88189
## 2004.4849       34.53557  -8.80989633  77.88104  -31.7556001 100.82674
## 2004.4877       34.28228  -9.18685478  77.75141  -32.1980243 100.76258
## 2004.4904       34.52241  -9.07004511  78.11486  -32.1464946 101.19131
## 2004.4932       35.11934  -8.59607915  78.83476  -31.7376244 101.97631
## 2004.4959       35.21505  -8.62299271  79.05309  -31.8294512 102.25955
## 2004.4986       34.52594  -9.43438535  78.48627  -32.7055760 101.75746
## 2004.5014       34.36932  -9.71294538  78.45159  -33.0486886 101.78733
## 2004.5041       34.58571  -9.61816856  78.78958  -33.0182863 102.18970
## 2004.5068       34.33396  -9.99118674  78.65911  -33.4555024 102.12342
## 2004.5096       34.22226 -10.22382831  78.66835  -33.7521667 102.19669
## 2004.5123       34.56059 -10.00610961  79.12730  -33.5982970 102.71948
## 2004.5151       34.66400 -10.02298746  79.35099  -33.6788515 103.00686
## 2004.5178       34.70945 -10.09751084  79.51640  -33.8168806 103.23577
## 2004.5205       34.44202 -10.48458591  79.36862  -34.2672919 103.15132
## 2004.5233       34.73400 -10.31192374  79.77993  -34.1577976 103.62581
## 2004.5260       34.52637 -10.63856735  79.69131  -34.5474422 103.60019
## 2004.5288       34.51031 -10.77332778  79.79395  -34.7450381 103.76566
## 2004.5315       34.17055 -11.23148030  79.57257  -35.2658618 103.60696
## 2004.5342       33.86918 -11.65092632  79.38929  -35.7478160 103.48618
## 2004.5370       34.45938 -11.17850676  80.09726  -35.3377429 104.25650
## 2004.5397       35.04309 -10.71226325  80.79845  -34.9336854 105.01987
## 2004.5425       34.65573 -11.21679868  80.52826  -35.5002476 104.81170
## 2004.5452       33.70113 -12.28826497  79.69053  -36.6335825 104.03585
## 2004.5479       34.08637 -12.01960559  80.19235  -36.4266350 104.59938
## 2004.5507       33.22014 -13.00212194  79.44240  -37.4707076 103.91098
## 2004.5534       33.36215 -12.97609681  79.70040  -37.5060842 104.23039
## 2004.5562       33.56557 -12.88838177  80.01952  -37.4796176 104.61075
## 2004.5589       34.24965 -12.31971869  80.81901  -36.9720508 105.47134
## 2004.5616       34.49290 -12.19159533  81.17739  -36.9048727 105.89067
## 2004.5644       33.86433 -12.93500589  80.66367  -37.7090786 105.43774
## 2004.5671       34.21427 -12.69963567  81.12817  -37.5343548 105.96289
## 2004.5699       34.51754 -12.51064646  81.54573  -37.4058643 106.44094
## 2004.5726       34.45110 -12.69109025  81.59330  -37.6466602 106.54887
## 2004.5753       34.71092 -12.54500652  81.96685  -37.5607829 106.98262
## 2004.5781       34.52445 -12.84493428  81.89384  -37.9207726 106.96968
## 2004.5808       33.84145 -13.64112900  81.32402  -38.7768857 106.45978
## 2004.5836       33.84735 -13.74814483  81.44284  -38.9436775 106.63838
## 2004.5863       34.15946 -13.54868283  81.86761  -38.8038499 107.12278
## 2004.5890       34.48478 -13.33575285  82.30531  -38.6504139 107.61997
## 2004.5918       34.37644 -13.55622103  82.30909  -38.9302365 107.68311
## 2004.5945       34.42376 -13.62075525  82.46828  -39.0539866 107.90151
## 2004.5973       34.22886 -13.92726106  82.38498  -39.4195708 107.87729
## 2004.6000       34.14968 -14.11778102  82.41715  -39.6690326 107.96840
## 2004.6027       33.55321 -14.82534495  81.93176  -40.4354026 107.54181
## 2004.6055       33.01051 -15.47887038  81.49990  -41.1475995 107.16863
## 2004.6082       32.69943 -15.90053249  81.29939  -41.6277992 107.02666
## 2004.6110       32.50444 -16.20585484  81.21473  -41.9915262 107.00040
## 2004.6137       33.27646 -15.54391130  82.09683  -41.3878554 107.94078
## 2004.6164       33.05548 -15.87472155  81.98569  -41.7768073 107.88777
## 2004.6192       32.47215 -16.56764302  81.51194  -42.5277402 107.47204
## 2004.6219       31.68214 -17.46698946  80.83128  -43.4849687 106.84926
## 2004.6247       31.49242 -17.76581148  80.75065  -43.8415443 106.82638
## 2004.6274       32.16547 -17.20161578  81.53256  -43.3349745 107.66592
## 2004.6301       32.45148 -17.02422549  81.92719  -43.2150834 108.11805
## 2004.6329       32.14579 -17.43829611  81.72988  -43.6865272 107.97811
## 2004.6356       32.77993 -16.91230318  82.47216  -43.2177824 108.77764
## 2004.6384       32.81377 -16.98637695  82.61391  -43.3489799 108.97651
## 2004.6411       33.25643 -16.65138908  83.16425  -43.0709922 109.58385
## 2004.6438       33.41547 -16.59979407  83.43073  -43.0762748 109.90721
## 2004.6466       32.65538 -17.46709931  82.77785  -44.0003356 109.31109
## 2004.6493       32.34972 -17.87973996  82.57918  -44.4696107 109.16905
## 2004.6521       32.17493 -18.16128959  82.51115  -44.8076744 109.15753
## 2004.6548       31.62662 -18.81613016  82.06937  -45.5189095 108.77215
## 2004.6575       31.31377 -19.23528501  81.86283  -45.9943400 108.62189
## 2004.6603       31.70802 -18.94712183  82.36316  -45.7623343 109.17837
## 2004.6630       31.14760 -19.61340297  81.90861  -46.4846556 108.77986
## 2004.6658       30.99661 -19.87003576  81.86326  -46.7972119 108.79043
## 2004.6685       31.20398 -19.76808770  82.17605  -46.7510715 109.15904
## 2004.6712       31.44393 -19.63334421  82.52121  -46.6720205 109.55988
## 2004.6740       30.55986 -20.62240918  81.74212  -47.7166634 108.83637
## 2004.6767       29.84914 -21.43790029  81.13618  -48.5876187 108.28590
## 2004.6795       30.51220 -20.87939984  81.90380  -48.0844694 109.10887
## 2004.6822       30.88390 -20.61205139  82.37984  -47.8723597 109.64015
## 2004.6849       30.75301 -20.84707512  82.35310  -48.1625104 109.66853
## 2004.6877       29.30193 -22.40208682  81.00594  -49.7725381 108.37639
## 2004.6904       29.30359 -22.50414016  81.11132  -49.9294971 108.53668
## 2004.6932       28.93462 -22.97662306  80.84587  -50.4567759 108.32602
## 2004.6959       28.63842 -23.37613030  80.65297  -50.9109700 108.18781
## 2004.6986       27.24644 -24.87121248  79.36409  -52.4606307 106.95351
## 2004.7014       27.21761 -25.00294077  79.43816  -52.6468297 107.08205
## 2004.7041       28.01493 -24.30831848  80.33817  -52.0065710 108.03642
## 2004.7068       28.33581 -24.08992473  80.76155  -51.8424344 108.51406
## 2004.7096       28.25290 -24.27513601  80.78093  -52.0817969 108.58759
## 2004.7123       27.58470 -25.04542811  80.21483  -52.9061350 108.07554
## 2004.7151       27.67943 -25.05259146  80.41146  -52.9672397 108.32611
## 2004.7178       29.08095 -23.75277957  81.91467  -51.7212652 109.88316
## 2004.7205       29.79034 -23.14488842  82.72558  -51.1671079 110.74780
## 2004.7233       29.90540 -23.13114288  82.94195  -51.2069934 111.01780
## 2004.7260       29.60713 -23.53053396  82.74479  -51.6599133 110.87417
## 2004.7288       26.43609 -26.80249465  79.67468  -54.9853011 107.85749
## 2004.7315       25.38571 -27.95361512  78.72503  -56.1897476 106.96117
## 2004.7342       26.23353 -27.20633615  79.67340  -55.4956942 107.96276
## 2004.7370       25.96302 -27.57721005  79.50324  -55.9196936 107.84573
## 2004.7397       27.14875 -26.49164405  80.78915  -54.8871538 109.18466
## 2004.7425       27.30707 -26.43330439  81.04745  -54.8817415 109.49589
## 2004.7452       28.03361 -25.80655882  81.87379  -54.3078249 110.37505
## 2004.7479       26.86605 -27.07373376  80.80584  -55.6277312 109.35983
## 2004.7507       27.11628 -26.92293781  81.15549  -55.5295693 109.76212
## 2004.7534       27.69460 -26.44385839  81.83306  -55.1030274 110.49223
## 2004.7562       26.46162 -27.77589991  80.69915  -56.4875102 109.41076
## 2004.7589       24.06916 -30.26724245  78.40557  -59.0311984 107.16953
## 2004.7616       23.34782 -31.08728819  77.78293  -59.9034948 106.59914
## 2004.7644       24.40858 -30.12505282  78.94222  -58.9934154 107.81058
## 2004.7671       25.29966 -29.33232365  79.93164  -58.2527482 108.85207
## 2004.7699       26.80589 -27.92426309  81.53604  -56.8966561 110.50844
## 2004.7726       28.42633 -26.40182019  83.25448  -55.4260886 112.27874
## 2004.7753       28.45967 -26.46630131  83.38564  -55.5423525 112.46169
## 2004.7781       27.98128 -27.04233826  83.00489  -56.1700802 112.13263
## 2004.7808       26.41156 -28.70952601  81.53265  -57.8888672 110.71199
## 2004.7836       26.16925 -29.04913402  81.38764  -58.2799833 110.61849
## 2004.7863       26.11669 -29.19883364  81.43220  -58.4811004 110.71447
## 2004.7890       24.65721 -30.75526519  80.06969  -60.0888593 109.40329
## 2004.7918       24.14158 -31.36768819  79.65085  -60.7525201 109.03568
## 2004.7945       24.02966 -31.57623340  79.63555  -61.0122138 109.07153
## 2004.7973       24.26027 -31.44207515  79.96261  -60.9291154 109.44965
## 2004.8000       26.01220 -29.78642802  81.81084  -59.3244398 111.34885
## 2004.8027       27.12352 -28.77123718  83.01827  -58.3601327 112.60717
## 2004.8055       26.30034 -29.69037402  82.29105  -59.3300659 111.93074
## 2004.8082       25.94620 -30.14030095  82.03271  -59.8307023 111.72311
## 2004.8110       25.15128 -31.03085535  81.33341  -60.7718798 111.07443
## 2004.8137       25.16164 -31.11595530  81.43924  -60.9075167 111.23080
## 2004.8164       24.55142 -31.82148425  80.92432  -61.6634971 110.76634
## 2004.8192       24.17858 -32.28946884  80.64663  -62.1818479 110.53901
## 2004.8219       21.91674 -34.64629400  78.47977  -64.5889546 108.42243
## 2004.8247       22.38637 -34.27148807  79.04423  -64.2643460 109.03708
## 2004.8274       23.31825 -33.43427049  80.07078  -63.4772418 110.11375
## 2004.8301       24.54557 -32.30146265  81.39260  -62.3944639 111.48560
## 2004.8329       21.92996 -35.01142329  78.87135  -65.1543714 109.01429
## 2004.8356       19.74903 -37.28654903  76.78461  -67.4793614 106.97742
## 2004.8384       22.37106 -34.75855665  79.50068  -65.0011511 109.74328
## 2004.8411       24.90459 -32.31891937  82.12809  -62.6112140 112.42039
## 2004.8438       22.39975 -34.91748894  79.71699  -65.2594024 110.05890
## 2004.8466       19.78895 -37.62187191  77.19976  -68.0133232 107.59122
## 2004.8493       20.16906 -37.33518426  77.67331  -67.7760928 108.11421
## 2004.8521       19.88838 -37.70913902  77.48590  -68.1994245 107.97619
## 2004.8548       20.54781 -37.14283842  78.23845  -67.6824210 108.77803
## 2004.8575       22.08421 -35.69941312  79.86783  -66.2882135 110.45663
## 2004.8603       24.79758 -33.07886568  82.67402  -63.7168046 113.31196
## 2004.8630       21.58510 -36.38402335  79.55422  -67.0710223 110.24122
## 2004.8658       19.51064 -38.55100972  77.57229  -69.2869903 108.30827
## 2004.8685       20.23679 -37.91723603  78.39083  -68.7021203 109.17571
## 2004.8712       19.88132 -38.36494411  78.12759  -69.1986545 108.96130
## 2004.8740       20.64938 -37.68896956  78.98774  -68.5714289 109.87020
## 2004.8767       19.93514 -38.49515820  78.36544  -69.4262897 109.29657
## 2004.8795       18.69222 -39.82987482  77.21432  -70.8096019 108.19405
## 2004.8822       19.87604 -38.73770907  78.48980  -69.7659557 109.51804
## 2004.8849       20.17560 -38.52966452  78.88087  -69.6063550 109.95756
## 2004.8877       20.09857 -38.69806847  78.89520  -69.8231273 110.02026
## 2004.8904       20.45215 -38.43571719  79.34001  -69.6090694 110.51336
## 2004.8932       21.63795 -37.34099768  80.61690  -68.5625685 111.83848
## 2004.8959       20.38060 -38.68929700  79.45050  -69.9590121 110.72021
## 2004.8986       19.90285 -39.25785926  79.06355  -70.5756446 110.38134
## 2004.9014       20.07142 -39.17995157  79.32279  -70.5457336 110.68858
## 2004.9041       19.48117 -39.86073548  78.82307  -71.2744407 110.23677
## 2004.9068       19.36476 -40.06753627  78.79705  -71.5290918 110.25861
## 2004.9096       21.76021 -37.76233736  81.28276  -69.2716705 112.79209
## 2004.9123       22.12976 -37.48290676  81.74242  -69.0399452 113.29946
## 2004.9151       18.56448 -41.13816826  78.26712  -72.7428399 109.87180
## 2004.9178       17.62145 -42.17103940  77.41394  -73.8232727 109.06618
## 2004.9205       18.14078 -41.74142648  78.02298  -73.4411500 109.72270
## 2004.9233       20.93677 -39.03501158  80.90855  -70.7821542 112.65569
## 2004.9260       18.99173 -41.06949386  79.05295  -72.8639850 110.84744
## 2004.9288       19.15340 -40.99713113  79.30394  -72.8389003 111.14571
## 2004.9315       16.36170 -43.87801636  76.60141  -75.7669935 108.49038
## 2004.9342       16.44368 -43.88507521  76.77244  -75.8211905 108.70856
## 2004.9370       17.20845 -43.20922285  77.62612  -75.1924069 109.60931
## 2004.9397       19.26835 -41.23811064  79.77480  -73.2682942 111.80499
## 2004.9425       19.16184 -41.43326716  79.75696  -73.5103814 111.83407
## 2004.9452       17.72538 -42.95825225  78.40902  -75.0822286 110.53300
## 2004.9479       18.03385 -42.73817743  78.80589  -74.9089476 110.97666
## 2004.9507       16.50794 -44.35235612  77.36824  -76.5698522 109.58574
## 2004.9534       16.33963 -44.60880873  77.28807  -76.8729630 109.55222
## 2004.9562       17.17020 -43.86625109  78.20665  -76.1769962 110.51739
## 2004.9589       16.35632 -44.76801605  77.48066  -77.1252849 109.83792
## 2004.9616       15.77002 -45.44207084  76.98212  -77.8457966 109.38585
## 2004.9644       15.32870 -45.97103216  76.62843  -78.4211484 109.07854
## 2004.9671       13.77887 -47.60836380  75.16611  -80.1048042 107.66255
## 2004.9699       14.25122 -47.22340486  75.72584  -79.7661036 108.26854
## 2004.9726       14.38582 -47.17605935  75.94770  -79.7649507 108.53659
## 2004.9753       15.38716 -46.26185866  77.03618  -78.8968772 109.67120
## 2004.9781       17.05263 -44.68340100  78.78866  -77.3644817 111.46974
## 2004.9808       18.41813 -43.40479135  80.24105  -76.1318693 112.96813
## 2004.9836       19.54626 -42.36343083  81.45595  -75.1364416 114.22896
## 2004.9863       15.81055 -46.18579092  77.80689  -79.0046701 110.62577
## 2004.9890       16.28706 -45.79580064  78.36993  -78.6604843 111.23461
## 2004.9918       17.03439 -45.13488541  79.20366  -78.0453097 112.11408
## 2004.9945       18.75844 -43.49711501  81.01400  -76.4532165 113.97010
## 2004.9973       20.30549 -42.03623758  82.64721  -75.0379531 115.64893
## 2005.0000       20.01323 -42.41454135  82.44100  -75.4618078 115.48827
## 2005.0027       20.16132 -42.35237980  82.67502  -75.4451346 115.76778
## 2005.0055       18.79990 -43.79960966  81.39942  -76.9377903 114.53760
## 2005.0082       18.06044 -44.62476717  80.74565  -77.8083116 113.92919
## 2005.0110       18.87994 -43.89084857  81.65072  -77.1196947 114.87957
## 2005.0137       18.71912 -44.13712056  81.57537  -77.4112068 114.84946
## 2005.0164       20.65887 -42.28271843  83.60046  -75.6019833 116.91973
## 2005.0192       17.99232 -45.03450358  81.01913  -78.3988860 114.38352
## 2005.0219       16.02033 -47.09159934  79.13227  -80.5010383 112.54170
## 2005.0247       20.11563 -43.08130195  83.31256  -76.5357368 116.76700
## 2005.0274       21.98155 -41.30026815  85.26337  -74.7996384 118.76274
## 2005.0301       21.76538 -41.60120354  85.13197  -75.1454490 118.67622
## 2005.0329       19.58277 -43.86847412  83.03402  -77.4575349 116.62308
## 2005.0356       18.75602 -44.77977587  82.29181  -78.4135922 115.92562
## 2005.0384       19.47598 -44.14424426  83.09620  -77.8227567 116.77472
## 2005.0411       20.18772 -43.51682783  83.89226  -77.2399771 117.61541
## 2005.0438       18.85494 -44.93381556  82.64369  -78.7015427 116.41142
## 2005.0466       19.79636 -44.07649318  83.66921  -77.8887394 117.48146
## 2005.0493       18.03219 -45.92464660  81.98904  -79.7813534 115.84574
## 2005.0521       16.92514 -47.11557984  80.96586  -81.0166888 114.86697
## 2005.0548       18.45992 -45.66456943  82.58441  -79.6100226 116.52986
## 2005.0575       20.17406 -44.03408851  84.38220  -78.0238280 118.37194
## 2005.0603       19.23714 -45.05455170  83.52884  -79.0885198 117.56281
## 2005.0630       19.73808 -44.63705708  84.11322  -78.7151965 118.19136
## 2005.0658       19.50550 -44.95296657  83.96398  -79.0752201 118.08623
## 2005.0685       17.90914 -46.63255399  82.45084  -80.7988646 116.61715
## 2005.0712       19.15798 -45.46683358  83.78280  -79.6771446 117.99311
## 2005.0740       21.87553 -42.83229568  86.58336  -77.0865506 120.83761
## 2005.0767       20.53740 -44.25333280  85.32813  -78.5514753 119.62628
## 2005.0795       20.19807 -44.67546500  85.07160  -79.0174390 119.41358
## 2005.0822       22.20058 -42.75564350  87.15681  -77.1413931 121.54256
## 2005.0849       23.09218 -41.94663885  88.13099  -76.3761084 122.56046
## 2005.0877       23.81598 -41.30531840  88.93728  -75.7784525 123.41042
## 2005.0904       23.59367 -41.61001090  88.79735  -76.1267542 123.31409
## 2005.0932       20.37457 -44.91138459  85.66053  -79.4716822 120.22083
## 2005.0959       18.87838 -46.48975329  84.24650  -81.0935503 118.85030
## 2005.0986       18.53407 -46.91612916  83.98427  -81.5633710 118.63151
## 2005.1014       19.23206 -46.30009946  84.76423  -80.9907317 119.45486
## 2005.1041       20.80432 -44.80970551  86.41835  -79.5436739 121.15232
## 2005.1068       22.68452 -43.01127120  88.38031  -77.7885218 123.15756
## 2005.1096       21.17052 -44.60692779  86.94797  -79.4274068 121.76845
## 2005.1123       21.41336 -44.44565274  87.27237  -79.3093066 122.13602
## 2005.1151       23.78461 -42.15586168  89.72507  -77.0626369 124.63185
## 2005.1178       22.95430 -43.06752992  88.97612  -78.0173733 123.92597
## 2005.1205       19.52718 -46.57590442  85.63026  -81.5687630 120.62312
## 2005.1233       20.44717 -45.73706968  86.63141  -80.7728906 121.66723
## 2005.1260       21.06967 -45.19562744  87.33497  -80.2743582 122.41370
## 2005.1288       20.29387 -46.05238432  86.64013  -81.1739724 121.76172
## 2005.1315       20.69749 -45.72963360  87.12461  -80.8940268 122.28900
## 2005.1342       19.67898 -46.82890340  86.18686  -82.0360496 121.39401
## 2005.1370       19.71554 -46.87300933  86.30408  -82.1228567 121.55393
## 2005.1397       21.06769 -45.60142389  87.73680  -80.8939208 123.02930
## 2005.1425       23.22396 -43.52561892  89.97355  -78.8607139 125.30864
## 2005.1452       23.90914 -42.92082019  90.73909  -78.2984620 126.11673
## 2005.1479       25.24340 -41.66683215  92.15363  -77.0869696 127.57377
## 2005.1507       24.71145 -42.27895902  91.70187  -77.7415412 127.16445
## 2005.1534       22.57550 -44.49499444  89.64600  -79.9999706 125.15097
## 2005.1562       22.35381 -44.79667436  89.50430  -80.3439940 125.05162
## 2005.1589       22.88516 -44.34522010  90.11554  -79.9348328 125.70515
## 2005.1616       22.18454 -45.12563403  89.49472  -80.7574896 125.12658
## 2005.1644       22.40008 -44.98980403  89.78996  -80.6638524 125.46401
## 2005.1671       25.00777 -42.46172287  92.47726  -78.1779142 128.19345
## 2005.1699       24.79341 -42.75560144  92.34241  -78.5138861 128.10070
## 2005.1726       24.51304 -43.11538986  92.14147  -78.9157184 127.94180
## 2005.1753       24.73004 -42.97772095  92.43780  -78.8200440 128.28012
## 2005.1781       23.01030 -44.77669336  90.79730  -80.6609618 126.68157
## 2005.1808       19.34047 -48.52567105  87.20661  -84.4518358 123.13278
## 2005.1836       19.77775 -48.16744526  87.72294  -84.1354577 123.69095
## 2005.1863       20.44623 -47.57791958  88.47039  -83.5877309 124.48020
## 2005.1890       22.58076 -45.52226140  90.68378  -81.5738233 126.73534
## 2005.1918       25.06703 -43.11476528  93.24883  -79.2080293 129.34210
## 2005.1945       24.72328 -43.53720887  92.98376  -79.6721270 129.11868
## 2005.1973       23.85013 -44.48895078  92.18921  -80.6654750 128.36574
## 2005.2000       23.35898 -45.05861083  91.77656  -81.2766934 127.99464
## 2005.2027       24.59989 -43.89611028  93.09589  -80.1557036 129.35549
## 2005.2055       26.31226 -42.26207267  94.88658  -78.5631292 131.18764
## 2005.2082       24.93436 -43.71820260  93.58693  -80.0606750 129.92940
## 2005.2110       23.10982 -45.62089358  91.84053  -82.0047348 128.22437
## 2005.2137       23.14327 -45.66550113  91.95204  -82.0906641 128.37720
## 2005.2164       24.91530 -43.97144226  93.80204  -80.4378802 130.26848
## 2005.2192       24.64446 -44.32016194  93.60908  -80.8278282 130.11675
## 2005.2219       24.37121 -44.67120945  93.41363  -81.2200575 129.96247
## 2005.2247       25.42344 -43.69668496  94.54356  -80.2866685 131.13355
## 2005.2274       25.97598 -43.22176196  95.17373  -79.8528347 131.80480
## 2005.2301       27.92252 -41.35275421  97.19780  -78.0248702 133.86991
## 2005.2329       27.16013 -42.19259417  96.51285  -78.9057074 133.22596
## 2005.2356       27.41139 -42.01869457  96.84147  -78.7727595 133.59553
## 2005.2384       28.75407 -40.75328851  98.26142  -77.5482594 135.05639
## 2005.2411       28.01375 -41.57078801  97.59830  -78.4066195 134.43413
## 2005.2438       27.64453 -42.01711566  97.30617  -78.8937625 134.18282
## 2005.2466       26.55936 -43.17930203  96.29802  -80.0967190 133.21544
## 2005.2493       28.07138 -41.74421375  97.88697  -78.7023560 134.84511
## 2005.2521       29.88792 -40.00451731  99.78036  -77.0033399 136.77918
## 2005.2548       29.35958 -40.60962563  99.32878  -77.6490840 136.36824
## 2005.2575       27.81721 -42.22867437  97.86309  -79.3087239 134.94314
## 2005.2603       27.63597 -42.48650975  97.75844  -79.6071061 134.87904
## 2005.2630       29.13788 -41.06110244  99.33687  -78.2222013 136.49797
## 2005.2658       29.67549 -40.59992736  99.95090  -77.8014847 137.15246
## 2005.2685       29.39689 -40.95486700  99.74865  -78.1968388 136.99062
## 2005.2712       28.35751 -42.07051420  98.78553  -79.3528567 136.06787
## 2005.2740       29.00005 -41.50415117  99.50425  -78.8268207 136.82692
## 2005.2767       27.59868 -42.98161888  98.17898  -80.3445719 135.54193
## 2005.2795       27.82218 -42.83412932  98.47850  -80.2373225 135.88169
## 2005.2822       29.22809 -41.50415572  99.96034  -78.9475458 137.40373
## 2005.2849       30.13013 -40.67796880 100.93823  -78.1615126 138.42177
## 2005.2877       30.72087 -40.16300489 101.60474  -77.6866595 139.12839
## 2005.2904       30.04487 -40.91469362 101.00443  -78.4784162 138.56815
## 2005.2932       30.32132 -40.71384937 101.35649  -78.3175972 138.96024
## 2005.2959       30.57247 -40.53823135 101.68317  -78.1819619 139.32690
## 2005.2986       30.29348 -40.89266493 101.47963  -78.5763358 139.16330
## 2005.3014       29.67076 -41.59075657 100.93228  -79.3143254 138.65585
## 2005.3041       29.88011 -41.45669681 101.21692  -79.2201215 138.98034
## 2005.3068       31.41306 -39.99895993 102.82508  -77.8021985 140.62831
## 2005.3096       31.96973 -39.51741988 103.45688  -77.3604304 141.29989
## 2005.3123       31.43152 -40.13067913 102.99372  -78.0134198 140.87646
## 2005.3151       31.07754 -40.55963727 102.71471  -78.4820665 140.63714
## 2005.3178       30.93141 -40.78065500 102.64348  -78.7427313 140.60556
## 2005.3205       31.02438 -40.76251075 102.81126  -78.7641928 140.81294
## 2005.3233       31.56732 -40.29430098 103.42895  -78.3355475 141.47020
## 2005.3260       31.30766 -40.62862623 103.24395  -78.7093961 141.32472
## 2005.3288       29.25900 -42.75186807 101.26987  -80.8721203 139.39013
## 2005.3315       29.20720 -42.87817531 101.29258  -81.0378691 139.45227
## 2005.3342       30.20564 -41.95417196 102.36544  -80.1532666 140.56454
## 2005.3370       32.03071 -40.20344644 104.26487  -78.4419012 142.50333
## 2005.3397       30.49587 -41.81256733 102.80431  -80.0903419 141.08208
## 2005.3425       30.30228 -42.08035703 102.68492  -80.3974110 141.00197
## 2005.3452       29.46330 -42.99346541 101.92006  -81.3497585 140.27635
## 2005.3479       30.11734 -42.41346993 102.64815  -80.8089621 141.04364
## 2005.3507       31.44472 -41.16006829 104.04950  -79.5947196 142.48415
## 2005.3534       30.93522 -41.74346748 103.61390  -80.2172380 142.08767
## 2005.3562       31.74299 -41.00951784 104.49549  -79.5223678 143.00834
## 2005.3589       32.45148 -40.37477379 105.27773  -78.9266637 143.82962
## 2005.3616       32.15963 -40.74029578 105.05956  -79.3311860 143.65045
## 2005.3644       31.91213 -41.06139987 104.88565  -79.6912512 143.51550
## 2005.3671       32.14132 -40.90573117 105.18837  -79.5745042 143.85714
## 2005.3699       32.35224 -40.76825891 105.47275  -79.4759146 144.18040
## 2005.3726       32.18983 -41.00404553 105.38371  -79.7505448 144.13021
## 2005.3753       32.69823 -40.56895228 105.96541  -79.3542562 144.75072
## 2005.3781       32.14524 -41.19516992 105.48566  -80.0192398 144.30973
## 2005.3808       33.10437 -40.30920304 106.51794  -79.1720002 145.38074
## 2005.3836       33.69082 -39.79583427 107.17748  -78.6973201 146.07896
## 2005.3863       33.61506 -39.94460564 107.17473  -78.8847417 146.11487
## 2005.3890       32.89056 -40.74205069 106.52316  -79.7207987 145.50191
## 2005.3918       31.33293 -42.37254138 105.03841  -81.3898631 144.05573
## 2005.3945       31.81424 -41.96403420 105.59251  -81.0198915 144.64836
## 2005.3973       31.79357 -42.05741988 105.64457  -81.1517749 144.73892
## 2005.4000       32.67250 -41.25114367 106.59615  -80.3839584 145.72896
## 2005.4027       32.57865 -41.41757500 106.57488  -80.5888117 145.74612
## 2005.4055       33.75419 -40.31455036 107.82292  -79.5241715 147.03255
## 2005.4082       34.36927 -39.77190226 108.51045  -79.0198702 147.75842
## 2005.4110       34.47923 -39.73431275 108.69278  -79.0205900 147.97905
## 2005.4137       34.61665 -39.66918862 108.90250  -78.9937380 148.22704
## 2005.4164       34.80720 -39.55086809 109.16527  -78.9136523 148.52805
## 2005.4192       33.55523 -40.87499481 107.98546  -80.2759768 147.38644
## 2005.4219       33.99493 -40.50738678 108.49724  -79.9465295 147.93638
## 2005.4247       32.81941 -41.75492378 107.39374  -81.2321903 146.87101
## 2005.4274       32.51460 -42.13167816 107.16088  -81.6470318 146.67624
## 2005.4301       32.94550 -41.77265443 107.66366  -81.3260585 147.21707
## 2005.4329       33.28041 -41.50955919 108.07038  -81.1009771 147.66180
## 2005.4356       33.92599 -40.93571802 108.78770  -80.5651133 148.41710
## 2005.4384       33.27267 -41.66071257 108.20605  -81.3280488 147.87339
## 2005.4411       33.49552 -41.50946089 108.50051  -81.2147019 148.20575
## 2005.4438       33.17568 -41.90084180 108.25220  -81.6439514 147.99531
## 2005.4466       32.94135 -42.20664169 108.08934  -81.9875839 147.87028
## 2005.4493       33.06158 -42.15780430 108.28097  -81.9765431 148.09971
## 2005.4521       33.92229 -41.36842596 109.21301  -81.2249255 149.06951
## 2005.4548       32.91469 -42.44729009 108.27668  -82.3415147 148.17090
## 2005.4575       32.56952 -42.86365833 108.00270  -82.7955723 147.93462
## 2005.4603       33.65607 -41.84824488 109.16038  -81.8178127 149.12994
## 2005.4630       33.67842 -41.89694933 109.25380  -81.9041356 149.26098
## 2005.4658       33.45668 -42.18969207 109.10305  -82.2344614 149.14782
## 2005.4685       34.09710 -41.62020438 109.81439  -81.7025215 149.89671
## 2005.4712       34.21924 -41.56892273 110.00740  -81.6887526 150.12723
## 2005.4740       34.35679 -41.50217231 110.21575  -81.6594798 150.37305
## 2005.4767       34.02227 -41.90741843 109.95196  -82.1021686 150.14671
## 2005.4795       34.53089 -41.46946422 110.53125  -81.7016223 150.76341
## 2005.4822       34.78040 -41.29055939 110.85135  -81.5600906 151.12088
## 2005.4849       34.53557 -41.60591929 110.67706  -81.9127889 150.98393
## 2005.4877       34.28228 -41.92967885 110.49424  -82.2738524 150.83841
## 2005.4904       34.52241 -41.75995602 110.80477  -82.1413990 151.18621
## 2005.4932       35.11934 -41.23335985 111.47204  -81.6520379 151.89072
## 2005.4959       35.21505 -41.20792340 111.63803  -81.6638023 152.09390
## 2005.4986       34.52594 -41.96724352 111.01912  -82.4602890 151.51217
## 2005.5014       34.36932 -42.19400584 110.93265  -82.7241839 151.46283
## 2005.5041       34.58571 -42.04770350 111.21912  -82.6149801 151.78639
## 2005.5068       34.33396 -42.36946576 111.03739  -82.9738070 151.64173
## 2005.5096       34.22226 -42.55111846 110.99564  -83.1924906 151.63701
## 2005.5123       34.56059 -42.28267540 111.40386  -82.9610446 152.08223
## 2005.5151       34.66400 -42.24909094 111.57710  -82.9644237 152.29243
## 2005.5178       34.70945 -42.27341158 111.69230  -83.0256744 152.44456
## 2005.5205       34.44202 -42.61054110 111.49457  -83.3997004 152.28373
## 2005.5233       34.73400 -42.38818816 111.85620  -83.2142107 152.68222
## 2005.5260       34.52637 -42.66539345 111.71814  -83.5282460 152.58099
## 2005.5288       34.51031 -42.75096569 111.77159  -83.6506150 152.67124
## 2005.5315       34.17055 -43.16017786 111.50127  -84.0965909 152.43769
## 2005.5342       33.86918 -43.53092912 111.26929  -84.5040729 152.24244
## 2005.5370       34.45938 -43.01005817 111.92881  -84.0198998 152.93865
## 2005.5397       35.04309 -42.49560445 112.58179  -83.5421111 153.62830
## 2005.5425       34.65573 -42.95216868 112.26363  -84.0353076 153.34676
## 2005.5452       33.70113 -43.97590065 111.37817  -85.0956393 152.49791
## 2005.5479       34.08637 -43.65974171 111.83248  -84.8160475 152.98879
## 2005.5507       33.22014 -44.59499121 111.03526  -85.7878316 152.22810
## 2005.5534       33.36215 -44.52192987 111.24623  -85.7512726 152.47558
## 2005.5562       33.56557 -44.38740724 111.51854  -85.6532200 152.78436
## 2005.5589       34.24965 -43.77216319 112.27145  -85.0744137 153.57370
## 2005.5616       34.49290 -43.59768352 112.58348  -84.9363397 153.92213
## 2005.5644       33.86433 -44.29496048 112.02362  -85.6699903 153.39865
## 2005.5671       34.21427 -44.01367744 112.44221  -85.4250490 153.85358
## 2005.5699       34.51754 -43.77899430 112.81407  -85.2266757 154.26175
## 2005.5726       34.45110 -43.91396118 112.81617  -85.3979206 154.30013
## 2005.5753       34.71092 -43.72261571 113.14446  -85.2428215 154.66466
## 2005.5781       34.52445 -43.97749508 113.02640  -85.5339156 154.58282
## 2005.5808       33.84145 -44.72885296 112.41174  -86.3214567 154.00435
## 2005.5836       33.84735 -44.79124172 112.48594  -86.4199972 154.11470
## 2005.5863       34.15946 -44.54736065 112.86629  -86.2122365 154.53116
## 2005.5890       34.48478 -44.29021787 113.25978  -85.9911828 154.96074
## 2005.5918       34.37644 -44.46667782 113.21955  -86.2037007 154.95657
## 2005.5945       34.42376 -44.48740668 113.33493  -86.2604563 155.10798
## 2005.5973       34.22886 -44.75030832 113.20803  -86.5593537 155.01707
## 2005.6000       34.14968 -44.89742367 113.19679  -86.7424338 155.04180
## 2005.6027       33.55321 -45.56178090 112.66819  -87.4427249 154.54914
## 2005.6055       33.01051 -46.17229593 112.19332  -88.0891430 154.11017
## 2005.6082       32.69943 -46.55114236 111.95000  -88.5038618 153.90272
## 2005.6110       32.50444 -46.81384217 111.82272  -88.8024033 153.81128
## 2005.6137       33.27646 -46.10946767 112.66239  -88.1338399 154.68676
## 2005.6164       33.05548 -46.39803701 112.50900  -88.4581899 154.56916
## 2005.6192       32.47215 -47.04890611 111.99320  -89.1448093 154.08910
## 2005.6219       31.68214 -47.90638721 111.27067  -90.0380103 153.40230
## 2005.6247       31.49242 -48.16352945 111.14837  -90.3308421 153.31568
## 2005.6274       32.16547 -47.55783805 111.88879  -89.7608102 154.09176
## 2005.6301       32.45148 -47.33913471 112.24210  -89.5777362 154.48070
## 2005.6329       32.14579 -47.71207350 112.00366  -89.9862743 154.27786
## 2005.6356       32.77993 -47.14512854 112.70499  -89.4548987 155.01476
## 2005.6384       32.81377 -47.17842866 112.80596  -89.5237383 155.15127
## 2005.6411       33.25643 -46.80284417 113.31570  -89.1836635 155.69652
## 2005.6438       33.41547 -46.71082820 113.54177  -89.1271275 155.95806
## 2005.6466       32.65538 -47.53788678 112.84864  -89.9896364 155.30039
## 2005.6493       32.34972 -47.91045374 112.60990  -90.3976241 155.09707
## 2005.6521       32.17493 -48.15210133 112.50196  -90.6746629 155.02452
## 2005.6548       31.62662 -48.76721020 112.02045  -91.3251336 154.57838
## 2005.6575       31.31377 -49.14680241 111.77435  -91.7400582 154.36760
## 2005.6603       31.70802 -48.81924438 112.23529  -91.4478033 154.86384
## 2005.6630       31.14760 -49.44629718 111.74150  -92.1101301 154.40533
## 2005.6658       30.99661 -49.66386690 111.65709  -92.3629446 154.35617
## 2005.6685       31.20398 -49.52301980 111.93098  -92.2573132 154.66528
## 2005.6712       31.44393 -49.34954010 112.23740  -92.1190202 155.00688
## 2005.6740       30.55986 -50.30003046 111.41974  -93.1046684 154.22438
## 2005.6767       29.84914 -51.07710739 110.77538  -93.9168743 153.61515
## 2005.6795       30.51220 -50.48035199 111.50475  -93.3552191 154.37962
## 2005.6822       30.88390 -50.17490665 111.94270  -93.0848452 154.85264
## 2005.6849       30.75301 -50.37199041 111.87801  -93.3169718 154.82299
## 2005.6877       29.30193 -51.88921791 110.49307  -94.8692135 153.47307
## 2005.6904       29.30359 -51.95364169 110.56083  -94.9686230 153.57581
## 2005.6932       28.93462 -52.38864855 110.25789  -95.4385872 153.30783
## 2005.6959       28.63842 -52.75083218 110.02767  -95.8356998 153.11254
## 2005.6986       27.24644 -54.20874205 108.70162  -97.3285103 151.82139
## 2005.7014       27.21761 -54.30344828 108.73867  -97.4580890 151.89331
## 2005.7041       28.01493 -53.57195310 109.60181  -96.7614381 152.79129
## 2005.7068       28.33581 -53.31683456 109.98846  -96.5411357 153.21276
## 2005.7096       28.25290 -53.46546811 109.97126  -96.7245574 153.23035
## 2005.7123       27.58470 -54.19932849 109.36873  -97.4931780 152.66258
## 2005.7151       27.67943 -54.17020510 109.52907  -97.4987870 152.85766
## 2005.7178       29.08095 -52.83425046 110.99615  -96.1975368 154.35943
## 2005.7205       29.79034 -52.19035952 111.77105  -95.5883226 155.16901
## 2005.7233       29.90540 -52.14075616 111.95156  -95.5733683 155.38417
## 2005.7260       29.60713 -52.50443041 111.71869  -95.9716640 155.18592
## 2005.7288       26.43609 -55.74081427 108.61300  -99.2426418 152.11483
## 2005.7315       25.38571 -56.85649695 107.62791 -100.3928909 151.16431
## 2005.7342       26.23353 -56.07391827 108.54098  -99.6448512 152.11192
## 2005.7370       25.96302 -56.40962961 108.33566 -100.0150741 151.94111
## 2005.7397       27.14875 -55.28903724 109.58654  -98.9289661 153.22647
## 2005.7425       27.30707 -55.19580650 109.80995  -98.8701925 153.48434
## 2005.7452       28.03361 -54.53430419 110.60153  -98.2431201 154.31035
## 2005.7479       26.86605 -55.76685584 109.49896  -99.5100746 153.24218
## 2005.7507       27.11628 -55.58156915 109.81412  -99.3591637 153.59171
## 2005.7534       27.69460 -55.06813063 110.45733  -98.8800740 154.26928
## 2005.7562       26.46162 -56.36594384 109.28919 -100.2122092 153.13546
## 2005.7589       24.06916 -58.82318795 106.96152 -102.7037483 150.84208
## 2005.7616       23.34782 -59.60926429 106.30491 -103.5240930 150.21974
## 2005.7644       24.40858 -58.61318771 107.43035 -102.5622580 151.37942
## 2005.7671       25.29966 -57.78674465 108.38606 -101.7700298 152.36935
## 2005.7699       26.80589 -56.34509669 109.95688 -100.3625702 153.97435
## 2005.7726       28.42633 -54.78919203 111.64185  -98.8408273 155.69348
## 2005.7753       28.45967 -54.82033623 111.73967  -98.9061068 155.82544
## 2005.7781       27.98128 -55.36316026 111.32571  -99.4830398 155.44559
## 2005.7808       26.41156 -56.99725831 109.82038 -101.1512204 153.97434
## 2005.7836       26.16925 -57.30389900 109.64241 -101.4919174 153.83043
## 2005.7863       26.11669 -57.42075293 109.65412 -101.6428013 153.87617
## 2005.7890       24.65721 -58.94445961 108.25889 -103.2005119 152.51494
## 2005.7918       24.14158 -59.52427778 107.80744 -103.8143078 152.09747
## 2005.7945       24.02966 -59.70033744 107.75965 -104.0243192 152.08363
## 2005.7973       24.26027 -59.53381215 108.05435 -103.8917197 152.41226
## 2005.8000       26.01220 -57.84591573 109.87032 -102.2377230 154.26213
## 2005.8027       27.12352 -56.79859262 111.04563 -101.2242739 155.47131
## 2005.8055       26.30034 -57.68571343 110.28639 -102.1452428 154.74592
## 2005.8082       25.94620 -58.10373987 109.99614 -102.5970916 154.48950
## 2005.8110       25.15128 -58.96250857 109.26506 -103.4896570 153.79221
## 2005.8137       25.16164 -59.01593691 109.33922 -103.5768564 153.90014
## 2005.8164       24.55142 -59.68990759 108.79275 -104.2845726 153.38741
## 2005.8192       24.17858 -60.12644658 108.48361 -104.7548316 153.11199
## 2005.8219       21.91674 -62.45193808 106.28541 -107.1140176 150.94749
## 2005.8247       22.38637 -62.04590973 106.81865 -106.7416583 151.51440
## 2005.8274       23.31825 -61.17758031 107.81409 -105.9069727 152.54348
## 2005.8301       24.54557 -60.01377050 109.10491 -104.7767813 153.86792
## 2005.8329       21.92996 -62.69283837 106.55276 -107.4894425 151.34936
## 2005.8356       19.74903 -64.93717988 104.43524 -109.7673520 149.26541
## 2005.8384       22.37106 -62.37851113 107.12064 -107.2422262 151.98435
## 2005.8411       24.90459 -59.90830468 109.71748 -104.8055377 154.61471
## 2005.8438       22.39975 -62.47641163 107.27591 -107.4071375 152.20664
## 2005.8466       19.78895 -65.15043790 104.72833 -110.1146318 149.69252
## 2005.8493       20.16906 -64.83349880 105.17162 -109.8311357 150.16926
## 2005.8521       19.88838 -65.17730674 104.95407 -110.2083619 149.98512
## 2005.8548       20.54781 -64.58096331 105.67658 -109.6454119 150.74102
## 2005.8575       22.08421 -63.10759856 107.27601 -108.2054158 152.37383
## 2005.8603       24.79758 -60.45721441 110.05237 -105.5883757 155.18353
## 2005.8630       21.58510 -63.73263750 106.90283 -108.8971182 152.06731
## 2005.8658       19.51064 -65.86999083 104.89127 -111.0677664 150.08905
## 2005.8685       20.23679 -65.20668503 105.68027 -110.4377310 150.91132
## 2005.8712       19.88132 -65.62496131 105.38760 -110.8892531 150.65190
## 2005.8740       20.64938 -64.91965470 106.21842 -110.2171680 151.51594
## 2005.8767       19.93514 -65.69661042 105.56689 -111.0273209 150.89760
## 2005.8795       18.69222 -67.00219269 104.38664 -112.3660760 149.75052
## 2005.8822       19.87604 -65.88099056 105.63308 -111.2780225 151.03011
## 2005.8849       20.17560 -65.64400705 105.99521 -111.0741634 151.42537
## 2005.8877       20.09857 -65.78356887 105.98070 -111.2468255 151.44396
## 2005.8904       20.45215 -65.49247174 106.39677 -110.9888046 151.89310
## 2005.8932       21.63795 -64.36910211 107.64501 -109.8984871 153.17439
## 2005.8959       20.38060 -65.68884646 106.45005 -111.2512597 152.01246
## 2005.8986       19.90285 -66.22894838 106.03464 -111.8243659 151.63006
## 2005.9014       20.07142 -66.12267442 106.26552 -111.7510723 151.89391
## 2005.9041       19.48117 -66.77518558 105.73752 -112.4365401 151.39887
## 2005.9068       19.36476 -66.95380662 105.68332 -112.6480939 151.37761
## 2005.9096       21.76021 -64.62052042 108.14094 -110.3477168 153.86814
## 2005.9123       22.12976 -64.31309447 108.57261 -110.0731763 154.33269
## 2005.9151       18.56448 -67.94045204 105.06941 -113.7333957 150.86235
## 2005.9178       17.62145 -68.94551014 104.18842 -114.7712920 150.01420
## 2005.9205       18.14078 -68.48817456 104.76973 -114.3467712 150.62832
## 2005.9233       20.93677 -65.75412688 107.62766 -111.6455148 153.51905
## 2005.9260       18.99173 -67.76106574 105.74452 -113.6852215 151.66868
## 2005.9288       19.15340 -67.66124847 105.96805 -113.6181487 151.92495
## 2005.9315       16.36170 -70.51476752 103.23816 -116.5043890 149.22778
## 2005.9342       16.44368 -70.49454805 103.38191 -116.5168674 149.40423
## 2005.9370       17.20845 -69.79150478 104.20840 -115.8464988 150.26340
## 2005.9397       19.26835 -67.79328855 106.32998 -113.8809341 152.41763
## 2005.9425       19.16184 -67.96142747 106.28512 -114.0817015 152.40539
## 2005.9452       17.72538 -69.45948090 104.91025 -115.6123602 151.06313
## 2005.9479       18.03385 -69.21255989 105.28027 -115.3980216 151.46573
## 2005.9507       16.50794 -70.79997740 103.81586 -117.0179984 150.03388
## 2005.9534       16.33963 -71.02975335 103.70901 -117.2803108 149.95957
## 2005.9562       17.17020 -70.26060311 104.60100 -116.5436742 150.88407
## 2005.9589       16.35632 -71.13585910 103.84850 -117.4514209 150.16406
## 2005.9616       15.77002 -71.78348807 103.32354 -118.1315178 149.67157
## 2005.9644       15.32870 -72.28610627 102.94350 -118.6665813 149.32397
## 2005.9671       13.77887 -73.89717705 101.45492 -120.3100746 147.86782
## 2005.9699       14.25122 -73.48603906 101.98847 -119.9313365 148.43377
## 2005.9726       14.38582 -73.41259587 102.18424 -119.8902707 148.66191
## 2005.9753       15.38716 -72.47237843 103.24670 -118.9824080 149.75673
## 2005.9781       17.05263 -70.86798452 104.97324 -117.4103464 151.51561
## 2005.9808       18.41813 -69.56351869 106.39978 -116.1381905 152.97445
## 2005.9836       19.54626 -68.49638164 107.58890 -115.1033409 154.19586
## 2005.9863       15.81055 -72.29304441 103.91414 -118.9322688 150.55336
## 2005.9890       16.28706 -71.87743560 104.45156 -118.5489027 151.12303
## 2005.9918       17.03439 -71.19098021 105.25975 -117.8946679 151.96344
## 2005.9945       18.75844 -69.52774764 107.04463 -116.2636337 153.78052
## 2005.9973       20.30549 -68.04148557 108.65246 -114.8095478 155.42052
## 2006.0000       20.01323 -68.39448185 108.42094 -115.1946981 155.22116
## 2006.0027       20.16132 -68.30708954 108.62973 -115.1394377 155.46208
## 2006.0055       18.79990 -69.72916499 107.32897 -116.5936231 154.19343

##NOTE: AIC = 20992.54, BIC = 21009.62 , RMSE = 2.55584 Also, the headline of the plot below contains ETS(A,N,N) showing that the automated model characterized error of the “tsWSR_PK” SERIES as Additative (A), trend as None (A) and seasonality as None (N) which is equivalent to simple exponential model(ses) with only alpha parameter i.e level.

#Model 2 : ARIMA Model(AutoRegressive Moving Average)

#Changing to stationary set

train.sta_Var3 <- diff(train2, diff = 1)
test.sta_Var3 <- diff(test2, diff = 1)

#2.1 Auto Arima model without sesonality:

set.seed(301)
model_Var3 <- auto.arima(train.sta_Var3, seasonal = FALSE)
summary(model_Var3)
## Series: train.sta_Var3 
## ARIMA(2,0,2) with zero mean 
## 
## Coefficients:
##          ar1      ar2      ma1      ma2
##       0.6678  -0.2397  -0.7097  -0.1307
## s.e.  0.0635   0.0498   0.0649   0.0613
## 
## sigma^2 estimated as 6.55:  log likelihood=-5168.65
## AIC=10347.3   AICc=10347.33   BIC=10375.76
## 
## Training set error measures:
##                         ME     RMSE      MAE MPE MAPE     MASE         ACF1
## Training set -0.0005861283 2.557022 1.781308 NaN  Inf 0.613599 -3.50044e-05

##NOTE: AIC = 10347.3, BIC = 10375.76 , RMSE = 2.557022

tsdisplay(residuals(model_Var3), lag.max = 45, main = '(1,1,1) Model Residuals')

#2.2 Auto Arima model with sesonality:

set.seed(301)
model2_Var3 <- auto.arima(train.sta_Var3, seasonal= TRUE)
summary(model2_Var3)
## Series: train.sta_Var3 
## ARIMA(5,0,1) with zero mean 
## 
## Coefficients:
##          ar1      ar2     ar3     ar4      ar5      ma1
##       0.8030  -0.3648  0.0437  0.0029  -0.0287  -0.8449
## s.e.  0.0332   0.0274  0.0304  0.0277   0.0244   0.0257
## 
## sigma^2 estimated as 6.552:  log likelihood=-5167.89
## AIC=10349.78   AICc=10349.84   BIC=10389.63
## 
## Training set error measures:
##                         ME     RMSE      MAE MPE MAPE      MASE          ACF1
## Training set -0.0005409411 2.556135 1.778874 NaN  Inf 0.6127603 -0.0007242331

##NOTE: AIC = 10349.78, BIC =10389.63 , RMSE = 2.556135

tsdisplay(residuals(model2_Var3), lag.max = 20, main = 'Seasonal Model Residuals')

NOTE: Above Graphs shows serious lags at 14. #2.3 Manual Arima

set.seed(301)
model1_Var3 <- arima(train.sta_Var3, order = c(2,0,14))
summary(model1_Var3)
## 
## Call:
## arima(x = train.sta_Var3, order = c(2, 0, 14))
## 
## Coefficients:
##          ar1      ar2      ma1      ma2      ma3     ma4      ma5      ma6
##       0.3552  -0.3002  -0.3929  -0.0841  -0.1333  -0.087  -0.0391  -0.0388
## s.e.  0.2875   0.1964   0.2871   0.1975   0.1147   0.107   0.0578   0.0265
##          ma7      ma8     ma9    ma10     ma11    ma12     ma13     ma14
##       0.0653  -0.0673  0.0302  0.0497  -0.0241  0.0883  -0.0470  -0.0530
## s.e.  0.0242   0.0290  0.0292  0.0241   0.0297  0.0257   0.0299   0.0283
##       intercept
##         -0.0004
## s.e.     0.0153
## 
## sigma^2 estimated as 6.418:  log likelihood = -5148.33,  aic = 10332.66
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE         ACF1
## Training set 0.001016119 2.533326 1.770402 NaN  Inf 0.6922123 0.0001662007

##NOTE: AIC = 10332.66, RMSE = 2.533326

tsdisplay(residuals(model1_Var3), lag.max = 20, main = 'Seasonal Model Residuals')

Fit the observed with the predicted values for both models in the same plot.

par(mfrow = c(2,2))
fit_auto_train_Var3 %>% forecast(h=341) %>% autoplot() 

model1_Var3 %>% forecast(h=341) %>% autoplot() 

model2_Var3 %>% forecast(h=341) %>% autoplot() 

model_Var3 %>% forecast(h=341) %>% autoplot() 

COMMENT:
1. The plots includes 80% to 95% confidence interval.
2. For ARIMA models it plots the forecast but because of the nature of autoregressive process of lag, these forecast stabilizes quite smoothly.

Plot and comment on the residuals of the fitted data for both models in the same plot.

Residual Plot - to confirm no problem with this model

par(mfrow= c(2,2))
hist(fit_auto_train_Var3$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(fit_auto_train_Var3$residuals))

hist(model_Var3$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model_Var3$residuals))

hist(model1_Var3$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model1_Var3$residuals))

hist(model2_Var3$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model2_Var3$residuals))

OBSERVATION: The residuals for each model are normally distributed, which shows the good fit of the models. It is normally distrbuted

Testing the autocorrelation and partial autocorrelation of the residuals (using the plots and Ljung-Box Q statistic) up to an appropriate lag *

  1. ACF & PACF Plots
par(mfrow = c(4,2))
acf(fit_auto_train_Var3$residuals, main = 'Correlogram')
pacf(fit_auto_train_Var3$residuals, main = 'Partial Corelogram')

acf(model_Var3$residuals, main = 'Correlogram')
pacf(model_Var3$residuals, main = 'Partial Corelogram')

acf(model1_Var3$residuals, main = 'Correlogram')
pacf(model1_Var3$residuals, main = 'Partial Corelogram')

acf(model2_Var3$residuals, main = 'Correlogram')
pacf(model2_Var3$residuals, main = 'Partial Corelogram')

NOTE: 1. Residuals are the difference between actaual and fitted values 2. These blue lines are signficance bounds 3. This graph shows us the autocorelations for insample forecast errors 4. Do not exceed these significance bounds for lags 1 to end

  1. Ljunx-Box Test to check the autocorelations of the residuals
Box.test(model_Var3$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model_Var3$residuals
## X-squared = 50.734, df = 20, p-value = 0.0001737
Box.test(model1_Var3$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model1_Var3$residuals
## X-squared = 8.9669, df = 20, p-value = 0.9833
Box.test(model2_Var3$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model2_Var3$residuals
## X-squared = 49.345, df = 20, p-value = 0.0002747

OBSERVATIONS: For above performed Ljunx-Box Test on model_Var3,model1_Var3 and model2_Var3 output, we can say that p-value is more than significance level 0.05 for model1_Var3.
Hence there is not a statistical significance.
We conclude that there is a liitle evidence of non-zero auto-corelations in the insample forecast errors at lags 1 to 20 model1_Var3.
So we fail to rejects the null hypothesis. Hence, we conclude that there is no serial auto corelation present between lags. Hence, the model_Var3 seems to be good fit compared to other models.

Test and comment on the normality of the residuals.

Here, further testing the normality by plotting Normal Q-Q plots. From below plots we observed that most of the datapoints lies on the line. Hence, we could conclude that our model produces residuals that are normal

par(mfrow = c(2,2))
qqnorm(fit_auto_train_Var3$residuals)
qqnorm(model_Var3$residuals)
qqnorm(model1_Var3$residuals)
qqnorm(model2_Var3$residuals)

Validating The Model by testing the model performance with Holdout set

Here, choosing RMSE as evaluation parameter. The lower RMSE indicates a more accurate forecast.

Below, is the comparison of model. We are fitting the model to test set which is the holdout set. And checking the accuracy using RMSE(Root mean square error) evaluation parameter.

accuracy(forecast(fit_auto_train_Var3), test2) ["Test set", "RMSE"]
## [1] 6.973589

#o/p - 6.973589

accuracy(forecast(model1_Var3), test.sta_Var3) ["Test set", "RMSE"]
## [1] 2.481816

#O/P - 2.481816

accuracy(forecast(model2_Var3), test.sta_Var3) ["Test set", "RMSE"]
## [1] 2.479714

#o/p - 2.479714

accuracy(forecast(model_Var3), test.sta_Var3) ["Test set", "RMSE"]
## [1] 2.479376

#O/P - 2.479376

NOTE:- Here, the RMSE value is less of ARIMA models compared to exponential model.
The minimum RMSE is of model_Var3.

COMMENT ON BEST MODEL AND ITS PARAMETERS

We observed the best model to be ARIMA model i.e.“model_Var3”.

summary(model_Var3)
## Series: train.sta_Var3 
## ARIMA(2,0,2) with zero mean 
## 
## Coefficients:
##          ar1      ar2      ma1      ma2
##       0.6678  -0.2397  -0.7097  -0.1307
## s.e.  0.0635   0.0498   0.0649   0.0613
## 
## sigma^2 estimated as 6.55:  log likelihood=-5168.65
## AIC=10347.3   AICc=10347.33   BIC=10375.76
## 
## Training set error measures:
##                         ME     RMSE      MAE MPE MAPE     MASE         ACF1
## Training set -0.0005861283 2.557022 1.781308 NaN  Inf 0.613599 -3.50044e-05

Auto Arima Model(2,0,2) is the best model,which means

p = 2,
d = 0
q = 2

For the best model, 2 member of lag is used as a predictor. As we transformed the series to stationary so members of differences needed for stationarity is 0. 2 are the number of lagged forecasts errors used in the prediction equation.
AICc- Means AIC with correction is equaivalent to 10347.33.
AIC = 10347.33, RMSE = 2.557022 which is minimum as compared to other models.

4. Forecasting on T at 850 hpa level (or about 1500 m height) (T85)

#Checking the attributes of time series

attributes(tsT85)
## $tsp
## [1] 1998.00 2004.94  365.00
## 
## $class
## [1] "ts"

#Plotting the series

plot(tsT85, main = "T at 850 hpa level (or about 1500 m height)(tsT85)", col = "red")

From the plot, we observes that there is no trend but seems to have seasonlity.
By just looking, it doesnt seems that data is missing.

Checking the statistics and visualising the missing data in the series:

plotNA.distribution(tsT85)

NOTE: Here, Data is missing at random

As we see the data is missing randomly, so locf and NOCB does seems the good technique to implement here.

#Impute the missing values with “na_locf” function of “imputeTS package” and visualise the imputed values in the time series

tsT85.imp <- na_locf(tsT85)
plotNA.imputations(tsT85, tsT85.imp)

#Decomposition Of Time Series

decomp3 <- decompose(tsT85.imp)
plot(decomp3, col = "red")

OBSERVATIONS:
1)There seems to have trend.
2)It seems to have downward trend.
3)We observe some seasonality in the series. Seasonality describes cyclical effects due to the time of the year.

Let’s explore this seasonality pattern by plotting the decomp$seasonal components such that y-axis represents Seasonality Index and x-axis has month

par(mfrow=c(1,2))
plot(decomp3$seasonal, type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")
plot(decomp3$seasonal[1:365], type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")

OBSERVATIONS:
1) There is increase in the mid of months of a year.
2) There is a drop in peak T for starting and ending months of the year

#Stationarity check of time series
We could check the stationarity of time series by ADF and KPSS Tests:

  1. Augmented Dickey- Fuller(ADF) t-statistic test for unit root
  2. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) for level or trend stationarity
adf.test(tsT85.imp)
## Warning in adf.test(tsT85.imp): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  tsT85.imp
## Dickey-Fuller = -4.4455, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary

NOTE: Here p-value is less, so the series is stationary

kpss.test(tsT85.imp, null="Trend")
## 
##  KPSS Test for Trend Stationarity
## 
## data:  tsT85.imp
## KPSS Trend = 0.15198, Truncation lag parameter = 8, p-value = 0.04502

NOTE: Here p-value is less, so the series is not stationary

Plotting ACF to check the auto-corelations

acf(tsT85.imp,lag.max = length(tsT85.imp),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Here, we observes most of the data lies outside the blue-dotted lines means the significance level.
Now, we will take the difference to make our data stationary.

It seems difference of 1 is sufficient to make the series stationary.

tsT85.sta <- diff(tsT85.imp)
attributes(tsT85.sta)
## $tsp
## [1] 1998.003 2004.940  365.000
## 
## $class
## [1] "ts"

Hence, after differencing our series is now transformed to stationary time series.

#Visualising the stationary series

acf(tsT85.sta, lag.max = length(tsT85.sta),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

#CREATING MODELS AND FORECASTING: USING EXPONENTIAL SMOOTHING AND ARIMA TECHNIQUES FOR TIME SERIES FORECASTING

1.Train - Test Split of the data

train3 <- window(tsT85.imp, end = c(2004, 3))
test3 <- window(tsT85.imp, start = c(2004, 4))

#Model 1 : Auto Exponential Smoothening Model

set.seed(301)
fit_auto_train_Var4 <- forecast(train3)
summary(fit_auto_train_Var4)
## 
## Forecast method: STL +  ETS(A,N,N)
## 
## Model Information:
## ETS(A,N,N) 
## 
## Call:
##  ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend) 
## 
##   Smoothing parameters:
##     alpha = 0.8933 
## 
##   Initial states:
##     l = 11.8188 
## 
##   sigma:  2.0766
## 
##      AIC     AICc      BIC 
## 20079.90 20079.91 20096.98 
## 
## Error measures:
##                      ME     RMSE     MAE       MPE     MAPE      MASE
## Training set 0.00241468 2.075702 1.44361 -4.866176 27.31777 0.4473464
##                    ACF1
## Training set 0.02562371
## 
## Forecasts:
##           Point Forecast        Lo 80    Hi 80        Lo 95     Hi 95
## 2004.0082      11.100237   8.43890376 13.76157   7.03007890  15.17040
## 2004.0110      10.869974   7.30136062 14.43859   5.41225055  16.32770
## 2004.0137      12.370131   8.08208803 16.65817   5.81213512  18.92813
## 2004.0164      12.962118   8.05909656 17.86514   5.46359361  20.46064
## 2004.0192      10.732447   5.28341641 16.18148   2.39887400  19.06602
## 2004.0219      10.696135   4.75103263 16.64124   1.60388561  19.78838
## 2004.0247      10.808086   4.40523086 17.21094   1.01576428  20.60041
## 2004.0274      13.228469   6.39847236 20.05847   2.78289038  23.67405
## 2004.0301      12.859264   5.62730893 20.09122   1.79894343  23.91958
## 2004.0329      12.738680   5.12596149 20.35140   1.09603199  24.38133
## 2004.0356      11.767162   3.79183806 19.74249  -0.43004318  23.96437
## 2004.0384      12.672229   4.35008387 20.99437  -0.05539346  25.39985
## 2004.0411      12.935372   4.28029212 21.59045  -0.30143025  26.17217
## 2004.0438      14.177364   5.20169069 23.15304   0.45025624  27.90447
## 2004.0466      10.797635   1.51243078 20.08284  -3.40285951  24.99813
## 2004.0493      11.810703   2.22595799 21.39545  -2.84789932  26.46930
## 2004.0521      12.395595   2.52039205 22.27080  -2.70722475  27.49842
## 2004.0548      11.255144   1.09778496 21.41250  -4.27919623  26.78948
## 2004.0575      11.820608   1.38872131 22.25249  -4.13358583  27.77480
## 2004.0603      10.189616  -0.50975665 20.88899  -6.17366214  26.55289
## 2004.0630       9.907523  -1.05280901 20.86785  -6.85485820  26.66990
## 2004.0658      12.506252   1.29103136 23.72147  -4.64594802  29.65845
## 2004.0685      11.231639  -0.23280583 22.69608  -6.30171624  28.76499
## 2004.0712       9.151921  -2.55644404 20.86029  -8.75447784  27.05832
## 2004.0740      12.895457   0.94815189 24.84276  -5.37636963  31.16728
## 2004.0767      13.498790   1.31722966 25.68035  -5.13129898  32.12888
## 2004.0795      12.977617   0.56622225 25.38901  -6.00397341  31.95921
## 2004.0822      13.081694   0.44464409 25.71874  -6.24500616  32.40839
## 2004.0849      13.364708   0.50596219 26.22345  -6.30104668  33.03046
## 2004.0877      12.422666  -0.65401764 25.49935  -7.57639578  32.42173
## 2004.0904      14.367918   1.07687024 27.65897  -5.95898568  34.69482
## 2004.0932      13.545399   0.04338924 27.04741  -7.10414306  34.19494
## 2004.0959      12.674579  -1.03514700 26.38430  -8.29263743  33.64179
## 2004.0986      12.870449  -1.04389239 26.78479  -8.40969966  34.15060
## 2004.1014      12.596089  -1.51990159 26.71208  -8.99245577  34.18463
## 2004.1041      11.758253  -2.55654707 26.07305 -10.13434460  33.65085
## 2004.1068      12.536246  -1.97464009 27.04713  -9.65623919  34.72873
## 2004.1096      12.058714  -2.64564319 26.76307 -10.42965977  34.54709
## 2004.1123      10.818531  -4.07678469 25.71385 -11.96188859  33.59895
## 2004.1151      13.143482  -1.94037523 28.22734  -9.92528681  36.21225
## 2004.1178      14.085787  -1.18428324 29.35586  -9.26777026  37.43934
## 2004.1205      13.683871  -1.77016901 29.13791  -9.95104376  37.31879
## 2004.1233      11.651619  -3.98422644 27.28746 -12.26134314  35.56458
## 2004.1260      11.477994  -4.33756725 27.29356 -12.70981966  35.66581
## 2004.1288      11.616076  -4.37718159 27.60933 -12.84350073  36.07565
## 2004.1315      13.957130  -2.21187143 30.12613 -10.77122357  38.68548
## 2004.1342      13.114389  -3.22846588 29.45724 -11.87985065  38.10863
## 2004.1370      10.389784  -6.12509410 26.90466 -14.86754272  35.64711
## 2004.1397      11.101115  -5.58401413 27.78624 -14.41658777  36.61882
## 2004.1425      12.268456  -4.58520314 29.12212 -13.50699145  38.04390
## 2004.1452      12.414861  -4.60566012 29.43538 -13.61577977  38.44550
## 2004.1479      13.675931  -3.50983185 30.86169 -12.60742523  39.95929
## 2004.1507      14.597821  -2.75160981 31.94725 -11.93584384  41.13149
## 2004.1534      13.834977  -3.67659289 31.34655 -12.94665784  40.61661
## 2004.1562      13.069035  -4.60318592 30.74126 -13.95829433  40.09636
## 2004.1589      11.800443  -6.03098100 29.63187 -15.47036672  39.07125
## 2004.1616      12.848860  -5.14035852 30.83808 -14.66327573  40.36100
## 2004.1644      12.930553  -5.21508883 31.07619 -14.82081116  40.68192
## 2004.1671      13.077464  -5.22326314 31.37819 -14.91108286  41.06601
## 2004.1699      14.620315  -3.83419401 33.07482 -13.60342122  42.84405
## 2004.1726      14.692896  -3.91412513 33.29992 -13.76408704  43.14988
## 2004.1753      14.270794  -4.48749851 33.02909 -14.41753874  42.95913
## 2004.1781      14.738626  -4.16972837 33.64698 -14.17920630  43.65646
## 2004.1808      11.857418  -7.19981560 30.91465 -17.28810573  41.00294
## 2004.1836       9.091919 -10.11304040 28.29688 -20.27953178  38.46337
## 2004.1863       9.914159  -9.43739808 29.26572 -19.68149377  39.50981
## 2004.1890      12.880308  -6.61674547 32.37736 -16.93786198  42.69848
## 2004.1918      13.535456  -6.10601483 33.17693 -16.50358163  43.57449
## 2004.1945      13.569903  -6.21493178 33.35474 -16.68839083  43.82820
## 2004.1973      13.771620  -6.15554796 33.69879 -16.70435330  44.24759
## 2004.2000      14.258552  -5.80993846 34.32704 -16.43355570  44.95066
## 2004.2027      14.964083  -5.24474229 35.17291 -15.94264829  45.87081
## 2004.2055      17.118255  -3.22993756 37.46645 -14.00161997  48.23813
## 2004.2082      16.635054  -3.85155771 37.12167 -14.69651467  47.96662
## 2004.2110      14.657036  -5.96706525 35.28114 -16.88480498  46.19888
## 2004.2137      13.368753  -7.39192727 34.12943 -18.38196777  45.11947
## 2004.2164      14.992072  -5.90429580 35.88844 -16.96616452  46.95031
## 2004.2192      15.548627  -5.48255178 36.57981 -16.61578532  47.71304
## 2004.2219      14.688821  -6.47631036 35.85395 -17.68045416  47.05810
## 2004.2247      13.925324  -7.37291753 35.22357 -18.64752562  46.49817
## 2004.2274      15.691895  -5.73863045 37.12242 -17.08326518  48.46705
## 2004.2301      16.003726  -5.55827125 37.56572 -16.97250299  48.97995
## 2004.2329      17.353855  -4.33881733 39.04653 -15.82222431  50.52993
## 2004.2356      16.781209  -5.04135616 38.60377 -16.59352415  50.15594
## 2004.2384      18.005325  -3.94636356 39.95701 -15.56688569  51.57754
## 2004.2411      17.256063  -4.82399528 39.33612 -16.51247183  51.02460
## 2004.2438      17.343729  -4.86395602 39.55141 -16.61999418  51.30745
## 2004.2466      16.668762  -5.66582062 39.00334 -17.48903434  50.82656
## 2004.2493      16.824957  -5.63580577 39.28572 -17.52581553  51.17573
## 2004.2521      18.180715  -4.40552392 40.76695 -16.36195656  52.72339
## 2004.2548      18.298618  -4.41240343 41.00964 -16.43489197  53.03213
## 2004.2575      17.518525  -5.31659756 40.35365 -17.40478105  52.44183
## 2004.2603      17.012144  -5.94640767 39.97070 -18.09993101  52.12422
## 2004.2630      17.713327  -5.36799476 40.79465 -17.58650853  53.01316
## 2004.2658      18.915166  -4.28827667 42.11861 -16.57143702  54.40177
## 2004.2685      17.588966  -5.73595696 40.91389 -18.08342543  53.26136
## 2004.2712      17.762945  -5.68282953 41.20872 -18.09427292  53.62016
## 2004.2740      17.878965  -5.68704104 41.44497 -18.16213128  53.92006
## 2004.2767      16.172771  -7.51285733 39.85840 -20.05127133  52.39681
## 2004.2795      15.560278  -8.24436977 39.36493 -20.84578933  51.96635
## 2004.2822      16.136639  -7.78643695 40.05972 -20.45054862  52.72383
## 2004.2849      16.661509  -7.37941215 40.70243 -20.10590710  53.42893
## 2004.2877      18.613103  -5.54508847 42.77129 -18.33366240  55.55987
## 2004.2904      19.773956  -4.50093913 44.04885 -17.35129214  56.89920
## 2004.2932      20.181792  -4.20924775 44.57283 -17.12108425  57.48467
## 2004.2959      20.157741  -4.34889333 44.66438 -17.32192193  57.63740
## 2004.2986      19.572442  -5.04924431 44.19413 -18.08317774  57.22806
## 2004.3014      18.427225  -6.30897770 43.16343 -19.40353268  56.25798
## 2004.3041      17.703779  -7.14641350 42.55397 -20.30131067  55.70887
## 2004.3068      18.020680  -6.94298043 42.98434 -20.15794425  56.19931
## 2004.3096      18.990083  -6.08653289 44.06670 -19.36129157  57.34146
## 2004.3123      18.675156  -6.51390846 43.86422 -19.84819388  57.19851
## 2004.3151      17.776550  -7.52446372 43.07756 -20.91801131  56.47111
## 2004.3178      17.508823  -7.90364660 42.92129 -21.35619529  56.37384
## 2004.3205      18.143298  -7.38013994 43.66674 -20.89143210  57.17803
## 2004.3233      18.356111  -7.27781626 43.99004 -20.84759757  57.55982
## 2004.3260      19.101355  -6.64258652 44.84530 -20.27060598  58.47332
## 2004.3288      18.540640  -7.31284748 44.39413 -20.99885727  58.08014
## 2004.3315      17.696061  -8.26651112 43.65863 -22.01026654  57.40239
## 2004.3342      17.992708  -8.07849131 44.06391 -21.87975076  57.86517
## 2004.3370      18.195672  -7.98370499 44.37505 -21.84222987  58.23357
## 2004.3397      20.025059  -6.26204987 46.31217 -20.17760452  60.22772
## 2004.3425      19.337963  -7.05643736 45.73236 -21.02878900  59.70472
## 2004.3452      18.636581  -7.86467728 45.13784 -21.89359598  59.16676
## 2004.3479      18.584852  -8.02283488 45.19254 -22.10809346  59.27780
## 2004.3507      19.183317  -7.53037415 45.89701 -21.67174816  60.03838
## 2004.3534      19.874027  -6.94524976 46.69330 -21.14251739  60.89057
## 2004.3562      19.796560  -7.12788874 46.72101 -21.38083081  60.97395
## 2004.3589      20.773392  -6.25581834 47.80260 -20.56421820  62.11100
## 2004.3616      20.159768  -6.97380093 47.29334 -21.33744448  61.65698
## 2004.3644      19.932499  -7.30502787 47.17003 -21.72370345  61.58870
## 2004.3671      20.076747  -7.26434271 47.41784 -21.73784107  61.89133
## 2004.3699      20.039185  -7.40507632 47.48345 -21.93319059  62.01156
## 2004.3726      20.056469  -7.49057731 47.60352 -22.07310293  62.18604
## 2004.3753      19.692924  -7.95652633 47.34237 -22.59326104  61.97911
## 2004.3781      19.807539  -7.94393654 47.55901 -22.63468030  62.24976
## 2004.3808      19.863035  -7.99009257 47.71616 -22.73464755  62.46072
## 2004.3836      21.277769  -6.67664032 49.23218 -21.47481085  64.03035
## 2004.3863      20.926291  -7.12903504 48.98162 -21.98062757  63.83321
## 2004.3890      21.059098  -7.09678248 49.21498 -22.00160552  64.11980
## 2004.3918      20.245526  -8.01055241 48.50160 -22.96841653  63.45947
## 2004.3945      19.612115  -8.74380541 47.96804 -23.75452320  62.97875
## 2004.3973      19.601465  -8.85394839 48.05688 -23.91733440  63.12026
## 2004.4000      19.975418  -8.57914104 48.52998 -23.69501175  63.64585
## 2004.4027      20.105230  -8.54813281 48.75859 -23.71630662  63.92677
## 2004.4055      20.201167  -8.55065862 48.95299 -23.77095580  64.17329
## 2004.4082      19.319197  -9.53075587 48.16915 -24.80299852  63.44139
## 2004.4110      20.635953  -8.31179504 49.58370 -23.63580707  64.90771
## 2004.4137      21.255782  -7.78943101 50.30100 -23.16503812  65.67660
## 2004.4164      21.229521  -7.91283130 50.37187 -23.33986094  65.79890
## 2004.4192      20.195248  -9.04392144 49.43442 -24.52220277  64.91270
## 2004.4219      20.377302  -8.95836476 49.71297 -24.48772863  65.24233
## 2004.4247      19.712661  -9.71918667 49.14451 -25.29946560  64.72479
## 2004.4274      19.661051  -9.86666449 49.18877 -25.49769264  64.81979
## 2004.4301      19.672858  -9.95041482 49.29613 -25.63202795  64.97774
## 2004.4329      19.799030  -9.91949281 49.51755 -25.65152827  65.24959
## 2004.4356      19.876861  -9.93660679 49.69033 -25.71890349  65.47263
## 2004.4384      20.734980  -9.17313273 50.64309 -25.00553111  66.47549
## 2004.4411      19.923246 -10.07921237 49.92570 -25.96155439  65.80805
## 2004.4438      19.890735 -10.20577285 49.98724 -26.13790194  65.91937
## 2004.4466      19.385694 -10.80457166 49.57596 -26.78633272  65.55772
## 2004.4493      19.706667 -10.57706467 49.99040 -26.60830405  66.02164
## 2004.4521      20.251052 -10.12585952 50.62796 -26.20642497  66.70853
## 2004.4548      20.365241 -10.10456431 50.83505 -26.23430500  66.96479
## 2004.4575      20.837822  -9.72459552 51.40024 -25.90336197  67.57901
## 2004.4603      20.355115 -10.29963469 51.00986 -26.52727880  67.23751
## 2004.4630      20.791813  -9.95499167 51.53862 -26.23136666  67.81499
## 2004.4658      21.191192  -9.64739186 52.02978 -25.97235226  68.35474
## 2004.4685      20.481799 -10.44829331 51.41189 -26.82169496  67.78529
## 2004.4712      20.739408 -10.28192152 51.76074 -26.70362152  68.18244
## 2004.4740      20.814072 -10.29822755 51.92637 -26.76808427  68.39623
## 2004.4767      21.115366 -10.08763858 52.31837 -26.60551161  68.83624
## 2004.4795      21.293893  -9.99955413 52.58734 -26.56530432  69.15309
## 2004.4822      21.384012  -9.99961671 52.76764 -26.61310606  69.38113
## 2004.4849      21.441909 -10.03164278 52.91546 -26.69273452  69.57655
## 2004.4877      21.297473 -10.26574508 52.86069 -26.97430358  69.56925
## 2004.4904      22.076523  -9.57610774 53.72915 -26.33199854  70.48505
## 2004.4932      22.195206  -9.54658599 53.93700 -26.34967576  70.74009
## 2004.4959      22.309310  -9.52139326 54.14001 -26.37154979  70.99017
## 2004.4986      22.291116  -9.62825074 54.21048 -26.52534293  71.10758
## 2004.5014      21.755836 -10.25194894 53.76362 -27.19584678  70.70752
## 2004.5041      22.035171 -10.06078828 54.13113 -27.05136283  71.12170
## 2004.5068      22.290169  -9.89372312 54.47406 -26.93084649  71.51118
## 2004.5096      21.530987 -10.74059793 53.80257 -27.82414329  70.88612
## 2004.5123      21.893317 -10.46572419 54.25236 -27.59556574  71.38220
## 2004.5151      22.143876 -10.30238515 54.59014 -27.47839810  71.76615
## 2004.5178      22.394683 -10.13856422 54.92793 -27.36062479  72.14999
## 2004.5205      22.093827 -10.52617342 54.71383 -27.79415881  71.98181
## 2004.5233      21.849357 -10.85716814 54.55588 -28.17095654  71.86967
## 2004.5260      21.190832 -11.60198873 53.98365 -28.96145930  71.34312
## 2004.5288      20.514867 -12.36402261 53.39376 -29.76905544  70.79879
## 2004.5315      21.043148 -11.92158580 54.00788 -29.37206193  71.45836
## 2004.5342      21.377299 -11.67305695 54.42765 -29.16885834  71.92346
## 2004.5370      21.305491 -11.83026477 54.44125 -29.37127431  71.98226
## 2004.5397      21.713465 -11.50747190 54.93440 -29.09357338  72.52050
## 2004.5425      22.009961 -11.29593815 55.31586 -28.92701623  72.94694
## 2004.5452      22.017628 -11.37301844 55.40827 -29.04895869  73.08421
## 2004.5479      21.492663 -11.98251545 54.96784 -29.70320430  72.68853
## 2004.5507      21.237449 -12.32204843 54.79695 -30.08737315  72.56227
## 2004.5534      20.897338 -12.74626683 54.54094 -30.55611557  72.35079
## 2004.5562      20.733311 -12.99419174 54.46081 -30.84845346  72.31508
## 2004.5589      21.173944 -12.63724872 54.98514 -30.53581320  72.88370
## 2004.5616      21.287553 -12.60712297 55.18223 -30.54988085  73.12499
## 2004.5644      21.741634 -12.23632082 55.71959 -30.22316350  73.70643
## 2004.5671      21.576427 -12.48460224 55.63746 -30.51542193  73.66828
## 2004.5699      21.894851 -12.24905047 56.03875 -30.32374019  74.11344
## 2004.5726      22.251280 -11.97529357 56.47785 -30.09374708  74.59631
## 2004.5753      21.906217 -12.40282896 56.21526 -30.56494081  74.37737
## 2004.5781      22.115503 -12.27581816 56.50682 -30.48148366  74.71249
## 2004.5808      22.023979 -12.44942020 56.49738 -30.69853541  74.74649
## 2004.5836      21.237686 -13.31759673 55.79297 -31.61005843  74.08543
## 2004.5863      21.576145 -13.06082780 56.21312 -31.39653353  74.54882
## 2004.5890      21.734275 -12.98419564 56.45274 -31.36304364  74.83159
## 2004.5918      22.001135 -12.79864259 56.80091 -31.22053183  75.22280
## 2004.5945      22.533583 -12.34731101 57.41448 -30.81214116  75.87931
## 2004.5973      22.749353 -12.21247069 57.71118 -30.72014213  76.21885
## 2004.6000      22.675904 -12.36666216 57.71847 -30.91707594  76.26888
## 2004.6027      22.027887 -13.09523535 57.15101 -31.68829321  75.74407
## 2004.6055      21.382274 -13.82122030 56.58577 -32.45682466  75.22137
## 2004.6082      20.633116 -14.65056787 55.91680 -33.32862182  74.59485
## 2004.6110      20.723915 -14.63977629 56.08761 -33.36018357  74.80801
## 2004.6137      21.165930 -14.27758796 56.60945 -33.04025297  75.37211
## 2004.6164      21.682840 -13.84032511 57.20601 -32.64515288  76.01083
## 2004.6192      21.741977 -13.86065748 57.34461 -32.70755369  76.19151
## 2004.6219      21.318824 -14.36310250 57.00075 -33.25197347  75.88962
## 2004.6247      20.329233 -15.43180952 56.09028 -34.36256217  75.02103
## 2004.6274      20.936372 -14.90361264 56.77636 -33.87615451  75.74890
## 2004.6301      21.981265 -13.93748793 57.90002 -32.95172719  76.91426
## 2004.6329      21.332651 -14.66469783 57.33000 -33.72054322  76.38585
## 2004.6356      21.078848 -14.99692500 57.15462 -34.09428589  76.25198
## 2004.6384      21.788416 -14.36561176 57.94244 -33.50439809  77.08123
## 2004.6411      21.821701 -14.41041277 58.05381 -33.59053508  77.23394
## 2004.6438      22.258517 -14.05151364 58.56855 -33.27288303  77.78992
## 2004.6466      22.189811 -14.19797113 58.57759 -33.46049928  77.84012
## 2004.6493      20.742659 -15.72270745 57.20803 -35.02630660  76.51163
## 2004.6521      20.844853 -15.69793368 57.38764 -35.04251662  76.73222
## 2004.6548      20.018382 -16.60166122 56.63843 -35.98714132  76.02391
## 2004.6575      20.533995 -16.16314254 57.23113 -35.58943369  76.65742
## 2004.6603      20.374670 -16.39939955 57.14874 -35.86641620  76.61576
## 2004.6630      20.189322 -16.66151975 57.04016 -36.16917688  76.54782
## 2004.6658      20.513849 -16.41360413 57.44130 -35.96181725  76.98952
## 2004.6685      20.345192 -16.65871520 57.34910 -36.24740034  76.93778
## 2004.6712      20.241127 -16.83907636 57.32133 -36.46815008  76.95040
## 2004.6740      19.755335 -17.40100668 56.91168 -37.07038603  76.58106
## 2004.6767      19.049696 -18.18262940 56.28202 -37.89223198  75.99162
## 2004.6795      19.229316 -18.07883845 56.53747 -37.82858232  76.28721
## 2004.6822      19.543842 -17.83998746 56.92767 -37.62979121  76.71747
## 2004.6849      19.571459 -17.88789223 57.03081 -37.71767492  76.86059
## 2004.6877      20.324329 -17.21039186 57.85905 -37.08007307  77.72873
## 2004.6904      18.867605 -18.74233572 56.47754 -38.65183547  76.38704
## 2004.6932      18.504384 -19.18062566 56.18939 -39.12986449  76.13863
## 2004.6959      19.234976 -18.52495258 56.99491 -38.51385148  76.98380
## 2004.6986      18.807392 -19.02730768 56.64209 -39.05578812  76.67057
## 2004.7014      18.841404 -19.06792038 56.75073 -39.13590429  76.81871
## 2004.7041      18.730012 -19.25378905 56.71381 -39.36119882  76.82122
## 2004.7068      18.593233 -19.46489987 56.65137 -39.61165833  76.79812
## 2004.7096      18.306417 -19.82590271 56.43874 -40.01193318  76.62477
## 2004.7123      17.458791 -20.74757044 55.66515 -40.97279665  75.89038
## 2004.7151      17.859130 -20.42113115 56.13939 -40.68547729  76.40374
## 2004.7178      18.141785 -20.21223316 56.49580 -40.51562387  76.79919
## 2004.7205      18.711876 -19.71575697 57.13951 -40.05811730  77.48187
## 2004.7233      18.225893 -20.27521467 56.72700 -40.65647010  77.10826
## 2004.7260      18.708023 -19.86641924 57.28247 -40.28649569  77.70254
## 2004.7288      18.945196 -19.70244190 57.59283 -40.16126571  78.05166
## 2004.7315      17.768960 -20.95173558 56.48965 -41.44923351  76.98715
## 2004.7342      17.730998 -21.06261687 56.52461 -41.59871607  77.06071
## 2004.7370      16.791329 -22.07506870 55.65773 -42.64969676  76.23235
## 2004.7397      16.936226 -22.00281808 55.87527 -42.61590298  76.48836
## 2004.7425      17.482168 -21.52938778 56.49372 -42.18085791  77.14519
## 2004.7452      18.189207 -20.89472531 57.27314 -41.58450946  77.96292
## 2004.7479      18.542392 -20.61378347 57.69857 -41.34181081  78.42660
## 2004.7507      17.627484 -21.60080148 56.85577 -42.36700158  77.62197
## 2004.7534      18.368852 -20.93141184 57.66912 -41.73571466  78.47342
## 2004.7562      16.596880 -22.77522985 55.96899 -43.61756574  76.81133
## 2004.7589      16.818755 -22.62507012 56.26258 -43.50536980  77.14288
## 2004.7616      16.522339 -22.99307083 56.03775 -43.91126540  76.95594
## 2004.7644      15.761034 -23.82583146 55.34790 -44.78185240  76.30392
## 2004.7671      16.854606 -22.80358722 56.51280 -43.79736637  77.50658
## 2004.7699      17.076627 -22.65276507 56.80602 -43.68423464  77.83749
## 2004.7726      16.584224 -23.21623952 56.38469 -44.28533208  77.45378
## 2004.7753      17.224579 -22.64682936 57.09599 -43.75347786  78.20264
## 2004.7781      17.092659 -22.84956836 57.03489 -43.99370609  78.17902
## 2004.7808      17.089652 -22.92326820 57.10257 -44.10482880  78.28413
## 2004.7836      17.660591 -22.42289798 57.74408 -43.64181546  78.96300
## 2004.7863      16.821719 -23.33221523 56.97565 -44.58842394  78.23186
## 2004.7890      16.114327 -24.10992802 56.33858 -45.40336264  77.63202
## 2004.7918      15.553872 -24.74058221 55.84833 -46.07117778  77.17892
## 2004.7945      16.219957 -24.14457405 56.58449 -45.51226594  77.95218
## 2004.7973      16.052219 -24.38226679 56.48671 -45.78699070  77.89143
## 2004.8000      15.216095 -25.28822490 55.72042 -46.72991689  77.16211
## 2004.8027      15.823018 -24.75101657 56.39705 -46.22961300  77.87565
## 2004.8055      16.069497 -24.57413234 56.71313 -46.08956992  78.22856
## 2004.8082      15.829887 -24.88321809 56.54299 -46.43543383  78.09521
## 2004.8110      15.771403 -25.01105851 56.55387 -46.59998977  78.14280
## 2004.8137      15.144422 -25.70727911 55.99612 -47.33286354  77.62171
## 2004.8164      14.861826 -26.05899767 55.78265 -47.72117326  77.44483
## 2004.8192      13.841905 -27.14792427 54.83173 -48.84662932  76.53044
## 2004.8219      13.244398 -27.81432090 54.30312 -49.54949401  76.03829
## 2004.8247      14.085482 -27.04201196 55.21298 -48.81359205  76.98456
## 2004.8274      15.127045 -26.06910853 56.32320 -47.87703482  78.13112
## 2004.8301      13.030070 -28.23462888 54.29477 -50.07884089  76.13898
## 2004.8329      13.924326 -27.40880484 55.25746 -49.28924240  77.13789
## 2004.8356      12.351511 -29.04993849 53.75296 -50.96654172  75.66956
## 2004.8384      13.189403 -28.28025245 54.65906 -50.23296177  76.61177
## 2004.8411      13.819218 -27.71853114 55.35697 -49.70728726  77.34572
## 2004.8438      14.453024 -27.15270750 56.05876 -49.17745143  78.08350
## 2004.8466      12.957081 -28.71652241 54.63068 -50.77719544  76.69136
## 2004.8493      11.276321 -30.46504361 53.01769 -52.56158732  75.11423
## 2004.8521      12.728505 -29.08051151 54.53752 -51.21286776  76.66988
## 2004.8548      11.828235 -30.04832354 53.70479 -52.21643448  75.87290
## 2004.8575      12.879551 -29.06444074 54.82354 -51.26824879  77.02735
## 2004.8603      12.938908 -29.07240942 54.95022 -51.31185728  77.18967
## 2004.8630      15.254812 -26.82372255 57.33335 -49.09875321  79.60838
## 2004.8658      13.756685 -28.38896017 55.90233 -50.69951687  78.21289
## 2004.8685      11.001470 -31.21117797 53.21412 -53.55720423  75.56015
## 2004.8712      11.759366 -30.52017972 54.03891 -52.90161933  76.42035
## 2004.8740      13.309972 -29.03636560 55.65631 -51.45316261  78.07311
## 2004.8767      13.619623 -28.79340121 56.03265 -51.24549994  78.48475
## 2004.8795      11.600660 -30.87894624 54.08027 -53.36629128  76.56761
## 2004.8822      11.791160 -30.75492380 54.33724 -53.27745999  76.85978
## 2004.8849      11.030053 -31.58240506 53.64251 -54.14007750  76.20018
## 2004.8877      12.956378 -29.72235026 55.63511 -52.31510430  78.22786
## 2004.8904      11.482914 -31.26198249 54.22781 -53.88976374  76.85559
## 2004.8932      12.090748 -30.72021391 54.90171 -53.38296824  77.56446
## 2004.8959      11.824955 -31.05197122 54.70188 -53.74964474  77.39955
## 2004.8986      10.653347 -32.28944124 53.59614 -55.02198031  76.32867
## 2004.9014      13.557841 -29.45070954 56.56639 -52.21806077  79.33374
## 2004.9041      12.455390 -30.61882132 55.52960 -53.42093156  78.33171
## 2004.9068      12.078343 -31.06142968 55.21812 -53.89824602  78.05493
## 2004.9096      11.930084 -31.27515063 55.13532 -54.14662042  78.00679
## 2004.9123      13.947313 -29.32328496 57.21791 -52.22935576  80.12398
## 2004.9151      13.652888 -29.68297427 56.98875 -52.62359389  79.92937
## 2004.9178      12.341342 -31.05968647 55.74237 -54.03480297  78.71749
## 2004.9205      14.066927 -29.39917008 57.53302 -52.40873174  80.54258
## 2004.9233      11.802496 -31.72857177 55.33356 -54.77252710  78.37752
## 2004.9260      11.102451 -32.49349166 54.69839 -55.57178939  77.77669
## 2004.9288      10.990515 -32.67020472 54.65124 -55.78279384  77.76382
## 2004.9315      10.633120 -33.09228204 54.35852 -56.23911174  77.50535
## 2004.9342      10.501696 -33.28829300 54.29168 -56.46931270  77.47270
## 2004.9370      12.321826 -31.53265384 56.17631 -54.74781319  79.39147
## 2004.9397      10.909636 -33.00924032 54.82851 -56.25848919  78.07776
## 2004.9425      10.954454 -33.02872500 54.93763 -56.31201348  78.22092
## 2004.9452      11.526065 -32.52132208 55.57345 -55.83860048  78.89073
## 2004.9479       9.657766 -34.45373606 53.76927 -57.80495490  77.12049
## 2004.9507       9.888240 -34.28728458 54.06376 -57.67239460  77.44887
## 2004.9534      10.834959 -33.40449466 55.07441 -56.82344682  78.49336
## 2004.9562       9.622613 -34.68067768 53.92590 -58.13342314  77.37865
## 2004.9589      10.959918 -33.40711829 55.32695 -56.89360843  78.81344
## 2004.9616      10.765645 -33.66504512 55.19633 -57.18523152  78.71652
## 2004.9644       9.846412 -34.64784043 54.34066 -58.20167489  77.89450
## 2004.9671       9.142303 -35.41542121 53.70003 -59.00285572  77.28746
## 2004.9699       8.515840 -36.10526649 53.13695 -59.72625327  76.75793
## 2004.9726       6.021963 -38.66243511 50.70636 -62.31692655  74.36085
## 2004.9753       8.058494 -36.68910601 52.80609 -60.37705473  76.49404
## 2004.9781       9.975409 -34.83530485 54.78612 -58.55666368  78.50748
## 2004.9808      10.739439 -34.13429911 55.61318 -57.88902103  79.36790
## 2004.9836       9.668891 -35.26778306 54.60556 -59.05582129  78.39360
## 2004.9863       9.141732 -35.85778986 54.14125 -59.67909781  77.96256
## 2004.9890       7.533067 -37.52921480 52.59535 -61.38374606  76.44988
## 2004.9918       9.660976 -35.46397914 54.78593 -59.35168751  78.67364
## 2004.9945       9.060576 -36.12696550 54.24812 -60.04780496  78.16896
## 2004.9973       9.798281 -35.45176029 55.04832 -59.40568502  79.00225
## 2005.0000      12.293157 -33.01929725 57.60561 -57.00626161  81.59258
## 2005.0027      12.709784 -32.66499814 58.08457 -56.68495669  82.10452
## 2005.0055      11.400790 -34.03623354 56.83781 -58.08914101  80.89072
## 2005.0082      11.100237 -34.39894340 56.59942 -58.48475473  80.68523
## 2005.0110      10.869974 -34.69127824 56.43123 -58.80994854  80.54990
## 2005.0137      12.370131 -33.25310918 57.99337 -57.40459374  82.14486
## 2005.0164      12.962118 -32.72302548 58.64726 -56.90727977  82.83152
## 2005.0192      10.732447 -35.01451683 56.47941 -59.23149651  80.69639
## 2005.0219      10.696135 -35.11256503 56.50483 -59.36222594  80.75450
## 2005.0247      10.808086 -35.06226747 56.67844 -59.34456562  80.96074
## 2005.0274      13.228469 -32.70345427 59.16039 -57.01834586  83.47528
## 2005.0301      12.859264 -33.13414810 58.85268 -57.48158949  83.20012
## 2005.0329      12.738680 -33.31613803 58.79350 -57.69608577  83.17344
## 2005.0356      11.767162 -34.34898019 57.88330 -58.76139099  82.29571
## 2005.0384      12.672229 -33.50515584 58.84961 -57.94998659  83.29444
## 2005.0411      12.935372 -33.30317441 59.17392 -57.78038218  83.65113
## 2005.0438      14.177364 -32.12226295 60.47699 -56.63180495  84.98653
## 2005.0466      10.797635 -35.56299216 57.15826 -60.10482580  81.70010
## 2005.0493      11.810703 -34.61084491 58.23225 -59.18492776  82.80633
## 2005.0521      12.395595 -34.08679263 58.87798 -58.69308242  83.48427
## 2005.0548      11.255144 -35.28800449 57.79829 -59.92645912  82.43675
## 2005.0575      11.820608 -34.78322251 58.42444 -59.45380005  83.09502
## 2005.0603      10.189616 -36.47481750 56.85405 -61.17747617  81.55671
## 2005.0630       9.907523 -36.81743431 56.63248 -61.55213250  81.36718
## 2005.0658      12.506252 -34.27915054 59.29166 -59.04584681  84.05835
## 2005.0685      11.231639 -35.61413190 58.07741 -60.41278496  82.87606
## 2005.0712       9.151921 -37.75414035 56.05798 -62.58470907  80.88855
## 2005.0740      12.895457 -34.07081617 59.86173 -58.93325958  84.72417
## 2005.0767      13.498790 -33.52761896 60.52520 -58.42189625  85.41948
## 2005.0795      12.977617 -34.10885073 60.06409 -59.03492124  84.99016
## 2005.0822      13.081694 -34.06475613 60.22814 -59.02257936  85.18597
## 2005.0849      13.364708 -33.84164841 60.57106 -58.83118402  85.56060
## 2005.0877      12.422666 -34.84352068 59.68885 -59.86472846  84.71006
## 2005.0904      14.367918 -32.95802271 61.69386 -58.01086264  86.74670
## 2005.0932      13.545399 -33.84022121 60.93102 -58.92465339  86.01545
## 2005.0959      12.674579 -34.77064571 60.11980 -59.88663041  85.23579
## 2005.0986      12.870449 -34.63430506 60.37520 -59.78180268  85.52270
## 2005.1014      12.596089 -34.96811935 60.16030 -60.14709046  85.33927
## 2005.1041      11.758253 -35.86533603 59.38184 -61.07574133  84.59225
## 2005.1068      12.536246 -35.14664986 60.21914 -60.38845021  85.46094
## 2005.1096      12.058714 -35.68341472 59.80084 -60.95657112  85.07400
## 2005.1123      10.818531 -36.98275732 58.61982 -62.28723090  83.92429
## 2005.1151      13.143482 -34.71689313 61.00386 -60.05264519  86.33961
## 2005.1178      14.085787 -33.83360132 62.00518 -59.20059330  87.37217
## 2005.1205      13.683871 -34.29445829 61.66220 -59.69265176  87.06039
## 2005.1233      11.651619 -36.38557901 59.68882 -61.81493569  85.11817
## 2005.1260      11.477994 -36.61800086 59.57399 -62.07848259  85.03447
## 2005.1288      11.616076 -36.53864376 59.77080 -62.03021256  85.26236
## 2005.1315      13.957130 -34.25624340 62.17050 -59.77886139  87.69312
## 2005.1342      13.114389 -35.15756612 61.38634 -60.71119558  86.93997
## 2005.1370      10.389784 -37.94068170 58.72025 -63.52528503  84.30485
## 2005.1397      11.101115 -37.28779179 59.49002 -62.90333155  85.10556
## 2005.1425      12.268456 -36.17882010 60.71573 -61.82525897  86.36217
## 2005.1452      12.414861 -36.09071478 60.92044 -61.76801557  86.59774
## 2005.1479      13.675931 -34.88787426 62.23974 -60.59599993  87.94786
## 2005.1507      14.597821 -34.02414398 63.21979 -59.76305760  88.95870
## 2005.1534      13.834977 -34.84507896 62.51503 -60.61474376  88.28470
## 2005.1562      13.069035 -35.66904216 61.80711 -61.46942149  87.60749
## 2005.1589      11.800443 -36.99558575 60.59647 -62.82664308  86.42753
## 2005.1616      12.848860 -36.00505191 61.70277 -61.86675086  87.56447
## 2005.1644      12.930553 -35.98117450 61.84228 -61.87347879  87.73458
## 2005.1671      13.077464 -35.89200978 62.04694 -61.81488329  87.96981
## 2005.1699      14.620315 -34.40683685 63.64747 -60.36024356  89.60087
## 2005.1726      14.692896 -34.39186732 63.77766 -60.37577137  89.76156
## 2005.1753      14.270794 -34.87151243 63.41310 -60.88587806  89.42747
## 2005.1781      14.738626 -34.46115691 63.93841 -60.50594849  89.98320
## 2005.1808      11.857418 -37.39977327 61.11461 -63.47495529  87.18979
## 2005.1836       9.091919 -40.22261447 58.40645 -66.32815156  84.51199
## 2005.1863       9.914159 -39.45764963 59.28597 -65.59350653  85.42183
## 2005.1890      12.880308 -36.54871036 62.30933 -62.71485196  88.47547
## 2005.1918      13.535456 -35.95070468 63.02162 -62.14709594  89.21801
## 2005.1945      13.569903 -35.97333480 63.11314 -62.19994085  89.33975
## 2005.1973      13.771620 -35.82862988 63.37187 -62.08541593  89.62866
## 2005.2000      14.258552 -35.39864326 63.91575 -61.68557467  90.20268
## 2005.2027      14.964083 -34.74999301 64.67816 -61.06703526  90.99520
## 2005.2055      17.118255 -32.65263701 66.88915 -58.99975567  93.23627
## 2005.2082      16.635054 -33.19258918 66.46270 -59.56974997  92.83986
## 2005.2110      14.657036 -35.22729316 64.54137 -61.63446189  90.94853
## 2005.2137      13.368753 -36.57219779 63.30970 -63.00934041  89.74685
## 2005.2164      14.992072 -35.00543747 64.98958 -61.47252004  91.45666
## 2005.2192      15.548627 -34.50537610 65.60263 -61.00236478  92.09962
## 2005.2219      14.688821 -35.42161230 64.79925 -61.94847337  91.32612
## 2005.2247      13.925324 -36.24147607 64.09212 -62.79817593  90.64882
## 2005.2274      15.691895 -34.53120910 65.91500 -61.11771426  92.50150
## 2005.2301      16.003726 -34.27561849 66.28307 -60.89189558  92.89935
## 2005.2329      17.353855 -32.98166713 67.68938 -59.62768289  94.33539
## 2005.2356      16.781209 -33.61042837 67.17285 -60.28614964  93.84857
## 2005.2384      18.005325 -32.44236434 68.45301 -59.14775807  95.15841
## 2005.2411      17.256063 -33.24761753 67.75974 -59.98265080  94.49478
## 2005.2438      17.343729 -33.21587974 67.90334 -59.98051971  94.66798
## 2005.2466      16.668762 -33.94671330 67.28424 -60.74092728  94.07845
## 2005.2493      16.824957 -33.84632275 67.49624 -60.67007812  94.31999
## 2005.2521      18.180715 -32.54630872 68.90774 -59.39957298  95.76100
## 2005.2548      18.298618 -32.48408808 69.08132 -59.36682884  95.96406
## 2005.2575      17.518525 -33.31980295 68.35685 -60.23198792  95.26904
## 2005.2603      17.012144 -33.88174381 67.90603 -60.82334082  94.84763
## 2005.2630      17.713327 -33.23606111 68.66272 -60.20703808  95.63369
## 2005.2658      18.915166 -32.08966241 69.91999 -59.08998737  96.92032
## 2005.2685      17.588966 -33.47124126 68.64917 -60.50088235  95.67881
## 2005.2712      17.762945 -33.35258181 68.87847 -60.41150727  95.93740
## 2005.2740      17.878965 -33.29182125 69.04975 -60.37999941  96.13793
## 2005.2767      16.172771 -35.05321615 67.39876 -62.17061547  94.51616
## 2005.2795      15.560278 -35.72084890 66.84141 -62.86743791  93.98799
## 2005.2822      16.136639 -35.19956930 67.47285 -62.37531665  94.64860
## 2005.2849      16.661509 -34.72972206 68.05274 -61.93459651  95.25761
## 2005.2877      18.613103 -32.83309194 70.05930 -60.06706233  97.29327
## 2005.2904      19.773956 -31.72714402 71.27506 -58.99017930  98.53809
## 2005.2932      20.181792 -31.37415396 71.73774 -58.66622317  99.02981
## 2005.2959      20.157741 -31.45299303 71.76848 -58.77406532  99.08955
## 2005.2986      19.572442 -32.09302209 71.23791 -59.44306671  98.58795
## 2005.3014      18.427225 -33.29291077 70.14736 -60.67189705  97.52635
## 2005.3041      17.703779 -34.07097184 69.47853 -61.47886923  96.88643
## 2005.3068      18.020680 -33.80862698 69.84999 -61.24540501  97.28677
## 2005.3096      18.990083 -32.89372371 70.87389 -60.35935201  98.33952
## 2005.3123      18.675156 -33.26309289 70.61341 -60.75754120  98.10785
## 2005.3151      17.776550 -34.21608452 69.76918 -61.73932264  97.29242
## 2005.3178      17.508823 -34.53814009 69.55579 -62.09013795  97.10778
## 2005.3205      18.143298 -33.95793618 70.24453 -61.53866379  97.82526
## 2005.3233      18.356111 -33.79933915 70.51156 -61.40876661  98.12099
## 2005.3260      19.101355 -33.10825396 71.31096 -60.74635148  98.94906
## 2005.3288      18.540640 -33.72307149 70.80435 -61.38980934  98.47109
## 2005.3315      17.696061 -34.62169795 70.01382 -62.31704653  97.70917
## 2005.3342      17.992708 -34.37904162 70.36446 -62.10297139  98.08839
## 2005.3370      18.195672 -34.23001390 70.62136 -61.98249544  98.37384
## 2005.3397      20.025059 -32.45450710 72.50462 -60.23551107 100.28563
## 2005.3425      19.337963 -33.19542738 71.87135 -61.00492451  99.68085
## 2005.3452      18.636581 -33.95057937 71.22374 -61.78854050  99.06170
## 2005.3479      18.584852 -34.05602326 71.22573 -61.92241932  99.09212
## 2005.3507      19.183317 -33.51121809 71.87785 -61.40602009  99.77265
## 2005.3534      19.874027 -32.87411364 72.62217 -60.79729269 100.54535
## 2005.3562      19.796560 -33.00513221 72.59825 -60.95665949 100.54978
## 2005.3589      20.773392 -32.08179634 73.62858 -60.06164314 101.60843
## 2005.3616      20.159768 -32.74886387 73.06840 -60.75700155 101.07654
## 2005.3644      19.932499 -33.02952166 72.89452 -61.06592167 100.93092
## 2005.3671      20.076747 -32.93860885 73.09210 -61.00324273 101.15674
## 2005.3699      20.039185 -33.02945200 73.10782 -61.12229138 101.20066
## 2005.3726      20.056469 -33.06539550 73.17833 -61.18641208 101.29935
## 2005.3753      19.692924 -33.48211585 72.86796 -61.63128143 101.01713
## 2005.3781      19.807539 -33.42062214 73.03570 -61.59790860 101.21299
## 2005.3808      19.863035 -33.41819499 73.14426 -61.62357430 101.34964
## 2005.3836      21.277769 -32.05647640 74.61201 -60.28992060 102.84546
## 2005.3863      20.926291 -32.46091778 74.31350 -60.72239899 102.57498
## 2005.3890      21.059098 -32.38102108 74.49922 -60.67051152 102.78871
## 2005.3918      20.245526 -33.24745238 73.73850 -61.56492435 102.05598
## 2005.3945      19.612115 -33.93366862 73.15790 -62.27909450 101.50333
## 2005.3973      19.601465 -33.99707313 73.20000 -62.37042538 101.57336
## 2005.4000      19.975418 -33.67582210 73.62666 -62.07707326 102.02791
## 2005.4027      20.105230 -33.59866154 73.80912 -62.02778423 102.23824
## 2005.4055      20.201167 -33.55532297 73.95766 -62.01228989 102.41462
## 2005.4082      19.319197 -34.48984047 73.12823 -62.97462440 101.61302
## 2005.4110      20.635953 -33.22558126 74.49749 -61.73815506 103.01006
## 2005.4137      21.255782 -32.65819701 75.16976 -61.19853362 103.71010
## 2005.4164      21.229521 -32.73685207 75.19589 -61.30492452 103.76397
## 2005.4192      20.195248 -33.82346889 74.21396 -62.41925027 102.80975
## 2005.4219      20.377302 -33.69370774 74.44831 -62.31717123 103.07178
## 2005.4247      19.712661 -34.41059103 73.83591 -63.06170988 102.48703
## 2005.4274      19.661051 -34.51439315 73.83649 -63.19314070 102.51524
## 2005.4301      19.672858 -34.55472779 73.90044 -63.26107744 102.60679
## 2005.4329      19.799030 -34.48064725 74.07871 -63.21457249 102.81263
## 2005.4356      19.876861 -34.45485708 74.20858 -63.21633147 102.97005
## 2005.4384      20.734980 -33.64873047 75.11869 -62.43772765 103.90769
## 2005.4411      19.923246 -34.51240647 74.35890 -63.32890015 103.17539
## 2005.4438      19.890735 -34.59680954 74.37828 -63.44077350 103.22224
## 2005.4466      19.385694 -35.15369455 73.92508 -64.02510266 102.79649
## 2005.4493      19.706667 -34.88451480 74.29785 -63.78334100 103.19668
## 2005.4521      20.251052 -34.39187537 74.89398 -63.31809368 103.82020
## 2005.4548      20.365241 -34.32938189 75.05986 -63.28296638 104.01345
## 2005.4575      20.837822 -33.90844835 75.58409 -62.88937318 104.56502
## 2005.4603      20.355115 -34.44275390 75.15298 -63.45099330 104.16122
## 2005.4630      20.791813 -34.05760598 75.64123 -63.09313426 104.67676
## 2005.4658      21.191192 -33.70972765 76.09211 -62.77251919 105.15490
## 2005.4685      20.481799 -34.47057466 75.43417 -63.56060391 104.52420
## 2005.4712      20.739408 -34.26437023 75.74319 -63.38161170 104.86043
## 2005.4740      20.814072 -34.24106318 75.86921 -63.38549147 105.01364
## 2005.4767      21.115366 -33.99107848 76.22181 -63.16266825 105.39340
## 2005.4795      21.293893 -33.86381349 76.45160 -63.06253947 105.65033
## 2005.4822      21.384012 -33.82490855 76.59293 -63.05074555 105.81877
## 2005.4849      21.441909 -33.81817805 76.70200 -63.07110094 105.95492
## 2005.4877      21.297473 -34.01373263 76.60868 -63.29371636 105.88866
## 2005.4904      22.076523 -33.28575438 77.43880 -62.59277396 106.74582
## 2005.4932      22.195206 -33.21809651 77.60851 -62.55212702 106.94254
## 2005.4959      22.309310 -33.15497048 77.77359 -62.51598707 107.13461
## 2005.4986      22.291116 -33.22409549 77.80633 -62.61207338 107.19431
## 2005.5014      21.755836 -33.81026016 77.32193 -63.22517464 106.73685
## 2005.5041      22.035171 -33.58176298 77.65211 -63.02358940 107.09393
## 2005.5068      22.290169 -33.37755644 77.95789 -62.84627023 107.42661
## 2005.5096      21.530987 -34.18748317 77.24946 -63.68305982 106.74503
## 2005.5123      21.893317 -33.87585282 77.66249 -63.39826789 107.18490
## 2005.5151      22.143876 -33.67594685 77.96370 -63.22517596 107.51293
## 2005.5178      22.394683 -33.47574691 78.26511 -63.05176574 107.84113
## 2005.5205      22.093827 -33.82716325 78.01482 -63.42994757 107.61760
## 2005.5233      21.849357 -34.12214955 77.82086 -63.75167517 107.45039
## 2005.5260      21.190832 -34.83114447 77.21281 -64.48738729 106.86905
## 2005.5288      20.514867 -35.55753374 76.58727 -65.24046971 106.27020
## 2005.5315      21.043148 -35.07963174 77.16593 -64.78923687 106.87553
## 2005.5342      21.377299 -34.79581549 77.55041 -64.53206586 107.28666
## 2005.5370      21.305491 -34.91791208 77.52889 -64.68078385 107.29177
## 2005.5397      21.713465 -34.56018258 77.98711 -64.34965195 107.77658
## 2005.5425      22.009961 -34.31388524 78.33381 -64.12992848 108.14985
## 2005.5452      22.017628 -34.35637342 78.39163 -64.19896687 108.23422
## 2005.5479      21.492663 -34.93144829 77.91677 -64.80056835 107.78589
## 2005.5507      21.237449 -35.23672758 77.71163 -65.13235071 107.60725
## 2005.5534      20.897338 -35.62685928 77.42154 -65.54896201 107.34364
## 2005.5562      20.733311 -35.84086300 77.30749 -65.78942192 107.25604
## 2005.5589      21.173944 -35.45016286 77.79805 -65.42515461 107.77304
## 2005.5616      21.287553 -35.38644264 77.96155 -65.38784394 107.96295
## 2005.5644      21.741634 -34.98220727 78.46547 -65.00999489 108.49326
## 2005.5671      21.576427 -35.19721530 78.35007 -65.25136607 108.40422
## 2005.5699      21.894851 -34.92854863 78.71825 -65.00903946 108.79874
## 2005.5726      22.251280 -34.62183395 79.12439 -64.72864178 109.23120
## 2005.5753      21.906217 -35.01656735 78.82900 -65.14966920 108.96210
## 2005.5781      22.115503 -34.85690903 79.08791 -65.01628198 109.24729
## 2005.5808      22.023979 -34.99801673 79.04597 -65.18363790 109.23160
## 2005.5836      21.237686 -35.83385078 78.30922 -66.04569738 108.52107
## 2005.5863      21.576145 -35.54488999 78.69718 -65.78293927 108.93523
## 2005.5890      21.734275 -35.43621532 78.90476 -65.70044460 109.16899
## 2005.5918      22.001135 -35.21876787 79.22104 -65.50915452 109.51142
## 2005.5945      22.533583 -34.73568879 79.80286 -65.05221023 110.11938
## 2005.5973      22.749353 -34.56924665 80.06795 -64.91188039 110.41059
## 2005.6000      22.675904 -34.69198079 80.04379 -65.06070436 110.41251
## 2005.6027      22.027887 -35.38923996 79.44501 -65.78403097 109.83981
## 2005.6055      21.382274 -36.08405302 78.84860 -66.50488913 109.26944
## 2005.6082      20.633116 -36.88236970 78.14860 -67.32922864 108.59546
## 2005.6110      20.723915 -36.84068708 78.28852 -67.31354662 108.76138
## 2005.6137      21.165930 -36.44774645 78.77961 -66.94658442 109.27844
## 2005.6164      21.682840 -35.97986890 79.34555 -66.50466321 109.87034
## 2005.6192      21.741977 -35.96972310 79.45368 -66.52045168 110.00441
## 2005.6219      21.318824 -36.44182540 79.07947 -67.01846625 109.65611
## 2005.6247      20.329233 -37.48032405 78.13879 -68.08285525 108.74132
## 2005.6274      20.936372 -36.92205211 78.79480 -67.55045175 109.42320
## 2005.6301      21.981265 -35.92598460 79.88851 -66.58023087 110.54276
## 2005.6329      21.332651 -36.62338292 79.28868 -67.30345403 109.96876
## 2005.6356      21.078848 -36.92592871 79.08363 -67.63180296 109.78950
## 2005.6384      21.788416 -36.26506328 79.84190 -66.99671899 110.57355
## 2005.6411      21.821701 -36.28044031 79.92384 -67.03785586 110.68126
## 2005.6438      22.258517 -35.89224438 80.40928 -66.67539824 111.19243
## 2005.6466      22.189811 -36.00953132 80.38915 -66.81840196 111.19802
## 2005.6493      20.742659 -37.50522233 78.99054 -68.33978832 109.82511
## 2005.6521      20.844853 -37.45152756 79.14123 -68.31176750 110.00147
## 2005.6548      20.018382 -38.32645748 78.36322 -69.21235003 109.24911
## 2005.6575      20.533995 -37.85926359 78.92725 -68.77078746 109.83878
## 2005.6603      20.374670 -38.06696690 78.81631 -69.00410086 109.75344
## 2005.6630      20.189322 -38.30065400 78.67930 -69.26337686 109.64202
## 2005.6658      20.513849 -38.02442496 79.05212 -69.01271559 110.04041
## 2005.6685      20.345192 -38.24134142 78.93173 -69.25517875 109.94556
## 2005.6712      20.241127 -38.39362589 78.87588 -69.43298888 109.91524
## 2005.6740      19.755335 -38.92759654 78.43827 -69.99246423 109.50314
## 2005.6767      19.049696 -39.68137579 77.78077 -70.77172725 108.87112
## 2005.6795      19.229316 -39.54985668 78.00849 -70.66567104 109.12430
## 2005.6822      19.543842 -39.28339202 78.37108 -70.42464846 109.51233
## 2005.6849      19.571459 -39.30379676 78.44671 -70.47047451 109.61339
## 2005.6877      20.324329 -38.59890918 79.24757 -69.79098752 110.43965
## 2005.6904      18.867605 -40.10357782 77.83879 -71.32103608 109.05625
## 2005.6932      18.504384 -40.51470374 77.52347 -71.75752131 108.76629
## 2005.6959      19.234976 -39.83197702 78.30193 -71.10013333 109.57009
## 2005.6986      18.807392 -40.30738809 77.92217 -71.60086262 109.21565
## 2005.7014      18.841404 -40.32116556 78.00397 -71.63993785 109.32275
## 2005.7041      18.730012 -40.48030704 77.94033 -71.82435667 109.28438
## 2005.7068      18.593233 -40.66479793 77.85126 -72.03410453 109.22057
## 2005.7096      18.306417 -40.99928737 77.61212 -72.39383062 109.00666
## 2005.7123      17.458791 -41.89454745 76.81213 -73.31430708 108.23189
## 2005.7151      17.859130 -41.54180552 77.26007 -72.98676131 108.70502
## 2005.7178      18.141785 -41.30670918 77.59028 -72.77684095 109.06041
## 2005.7205      18.711876 -40.78413818 78.20789 -72.27942582 109.70318
## 2005.7233      18.225893 -41.31760390 77.76939 -72.83802732 109.28981
## 2005.7260      18.708023 -40.88291861 78.29897 -72.42845779 109.84450
## 2005.7288      18.945196 -40.69315282 78.58355 -72.26378777 110.15418
## 2005.7315      17.768960 -41.91675876 77.45468 -73.51246956 109.05039
## 2005.7342      17.730998 -42.00205233 77.46405 -73.62281908 109.08482
## 2005.7370      16.791329 -42.98901577 76.57167 -74.63481864 108.21748
## 2005.7397      16.936226 -42.89137541 76.76383 -74.56219461 108.43465
## 2005.7425      17.482168 -42.39265336 77.35699 -74.08846915 109.05280
## 2005.7452      18.189207 -41.73279646 78.11121 -73.45358913 109.83200
## 2005.7479      18.542392 -41.42675684 78.51154 -73.17250674 110.25729
## 2005.7507      17.627484 -42.38877308 77.64374 -74.15946061 109.41443
## 2005.7534      18.368852 -41.69447704 78.43218 -73.49008264 110.22779
## 2005.7562      16.596880 -43.51348337 76.70724 -75.33398754 108.52775
## 2005.7589      16.818755 -43.33860606 76.97612 -75.18398931 108.82150
## 2005.7616      16.522339 -43.68198264 76.72666 -75.55222557 108.59690
## 2005.7644      15.761034 -44.49021200 76.01228 -76.38529522 107.90736
## 2005.7671      16.854606 -43.44352872 77.15274 -75.36343290 109.07264
## 2005.7699      17.076627 -43.26835915 77.42161 -75.21306501 109.36632
## 2005.7726      16.584224 -43.80757719 76.97602 -75.77706548 108.94551
## 2005.7753      17.224579 -43.21400106 77.66316 -75.20825259 109.65741
## 2005.7781      17.092659 -43.39266391 77.57798 -75.41165953 109.59698
## 2005.7808      17.089652 -43.44237686 77.62168 -75.48609745 109.66540
## 2005.7836      17.660591 -42.91810841 78.23929 -74.98653492 110.30772
## 2005.7863      16.821719 -43.80361552 77.44705 -75.89672893 109.54017
## 2005.7890      16.114327 -44.55760570 76.78626 -76.67538704 108.90404
## 2005.7918      15.553872 -45.16462424 76.27237 -77.30705457 108.41480
## 2005.7945      16.219957 -44.54506683 76.98498 -76.71212726 109.15204
## 2005.7973      16.052219 -44.75929618 76.86373 -76.95096786 109.05541
## 2005.8000      15.216095 -45.64187619 76.07407 -77.85814034 108.29033
## 2005.8027      15.823018 -45.08137453 76.72741 -77.32221237 108.96825
## 2005.8055      16.069497 -44.88128120 77.02027 -77.14667402 109.28567
## 2005.8082      15.829887 -45.16724152 76.82701 -77.45717065 109.11694
## 2005.8110      15.771403 -45.27203969 76.81485 -77.58648650 109.12929
## 2005.8137      15.144422 -45.94530066 76.23415 -78.28424657 108.57309
## 2005.8164      14.861826 -46.27414173 75.99779 -78.63756817 108.36122
## 2005.8192      13.841905 -47.34027245 75.02408 -79.72816093 107.41197
## 2005.8219      13.244398 -47.98395428 74.47275 -80.39628635 106.88508
## 2005.8247      14.085482 -47.18901115 75.35997 -79.62576838 107.79673
## 2005.8274      15.127045 -46.19355363 76.44764 -78.65471764 108.90881
## 2005.8301      13.030070 -48.33659949 74.39674 -80.82215194 106.88229
## 2005.8329      13.924326 -47.48838006 75.33703 -79.99830267 107.84695
## 2005.8356      12.351511 -49.10719695 73.81022 -81.64147145 106.34449
## 2005.8384      13.189403 -48.31527230 74.69408 -80.87388048 107.25269
## 2005.8411      13.819218 -47.73139004 75.36983 -80.31431373 107.95275
## 2005.8438      14.453024 -47.14348265 76.04953 -79.75070371 108.65675
## 2005.8466      12.957081 -48.68529053 74.59945 -81.31679087 107.23095
## 2005.8493      11.276321 -50.41188097 72.96452 -83.06764254 105.62028
## 2005.8521      12.728505 -49.00549390 74.46250 -81.68549869 107.14251
## 2005.8548      11.828235 -49.95152630 73.60800 -82.65575634 106.31223
## 2005.8575      12.879551 -48.94593875 74.70504 -81.67437611 107.43348
## 2005.8603      12.938908 -48.93227714 74.81009 -81.68490392 107.56272
## 2005.8630      15.254812 -46.66203397 77.17166 -79.43883233 109.94846
## 2005.8658      13.756685 -48.20578885 75.71916 -81.00674097 108.52011
## 2005.8685      11.001470 -51.00659702 73.00954 -83.83168513 105.83463
## 2005.8712      11.759366 -50.29426183 73.81299 -83.14346819 106.66220
## 2005.8740      13.309972 -48.78918302 75.40913 -81.66248995 108.28243
## 2005.8767      13.619623 -48.52502577 75.76427 -81.42241560 108.66166
## 2005.8795      11.600660 -50.58944936 73.79077 -83.51090448 106.71222
## 2005.8822      11.791160 -50.44437646 74.02670 -83.38987929 106.97220
## 2005.8849      11.030053 -51.25087784 73.31098 -84.22041083 106.28052
## 2005.8877      12.956378 -49.36991332 75.28267 -82.36345898 108.27622
## 2005.8904      11.482914 -50.88870558 73.85453 -83.90624643 106.87207
## 2005.8932      12.090748 -50.32616638 74.50766 -83.36768501 107.54918
## 2005.8959      11.824955 -50.63722202 74.28713 -83.70270105 107.35261
## 2005.8986      10.653347 -51.85405893 73.16075 -84.94348099 106.25018
## 2005.9014      13.557841 -48.99476228 76.11044 -82.10811007 109.22379
## 2005.9041      12.455390 -50.14237686 75.05316 -83.27963310 108.19041
## 2005.9068      12.078343 -50.56455540 74.72124 -83.72570286 107.88239
## 2005.9096      11.930084 -50.75791354 74.61808 -83.94293501 107.80310
## 2005.9123      13.947313 -48.78575165 76.68038 -81.99462997 109.88926
## 2005.9151      13.652888 -49.12521098 76.43099 -82.35792903 109.66370
## 2005.9178      12.341342 -50.48175907 75.16444 -83.73829975 108.42098
## 2005.9205      14.066927 -48.80114405 76.93500 -82.08149032 110.21534
## 2005.9233      11.802496 -51.11051223 74.71550 -84.41464706 108.01964
## 2005.9260      11.102451 -51.85546336 74.06036 -85.18336977 107.38827
## 2005.9288      10.990515 -52.01227205 73.99330 -85.36393311 107.34496
## 2005.9315      10.633120 -52.41450902 73.68075 -85.78990782 107.05615
## 2005.9342      10.501696 -52.59074330 73.59413 -85.98986297 106.99325
## 2005.9370      12.321826 -50.81539079 75.45904 -84.23821448 108.88187
## 2005.9397      10.909636 -52.27232688 74.09160 -85.71883780 107.53811
## 2005.9425      10.954454 -52.27222378 74.18113 -85.74240518 107.65131
## 2005.9452      11.526065 -51.74529536 74.79743 -85.23913049 108.29126
## 2005.9479       9.657766 -53.65824576 72.97378 -87.17571794 106.49125
## 2005.9507       9.888240 -53.47239229 73.24887 -87.01348486 106.78996
## 2005.9534      10.834959 -52.57026164 74.24018 -86.13495797 107.80488
## 2005.9562       9.622613 -53.82716484 73.07239 -87.41544835 106.66067
## 2005.9589      10.959918 -52.53438621 74.45422 -86.14624035 108.06608
## 2005.9616      10.765645 -52.77315405 74.30444 -86.40856230 107.93985
## 2005.9644       9.846412 -53.73685030 73.42967 -87.39579618 107.08862
## 2005.9671       9.142303 -54.48539162 72.77000 -88.16785868 106.45247
## 2005.9699       8.515840 -55.15625673 72.18794 -88.86222854 105.89391
## 2005.9726       6.021963 -57.69450412 69.73843 -91.42396433 103.46789
## 2005.9753       8.058494 -55.70231245 71.81930 -89.45524469 105.57223
## 2005.9781       9.975409 -53.82970706 73.78052 -87.60609503 107.55691
## 2005.9808      10.739439 -53.10995508 74.58883 -86.90978250 108.38866
## 2005.9836       9.668891 -54.22475052 73.56253 -88.04800114 107.38578
## 2005.9863       9.141732 -54.79612620 73.07959 -88.64278383 106.92625
## 2005.9890       7.533067 -56.44897712 71.51511 -90.31902557 105.38516
## 2005.9918       9.660976 -54.36522424 73.68718 -88.25864736 107.58060
## 2005.9945       9.060576 -55.00974986 73.13090 -88.92653156 107.04768
## 2005.9973       9.798281 -54.31614011 73.91270 -88.25626430 107.85283
## 2006.0000      12.293157 -51.86532844 76.45164 -85.82877907 110.41509
## 2006.0027      12.709784 -51.49273629 76.91230 -85.47949737 110.89906
## 2006.0055      11.400790 -52.84573398 75.64731 -86.85578952 109.65737

##NOTE: AIC = 20079.90 , BIC = 20096.98 , RMSE = 2.075702 Also, the headline of the plot below contains ETS(A,N,N) showing that the automated model characterized error of the “tsWSR_PK” SERIES as Additative (A), trend as None (A) and seasonality as None (N) which is equivalent to simple exponential model(ses) with only alpha parameter i.e level.

#Model 2 : ARIMA Models(AutoRegressive Moving Average) It is used for forecasting for both seasonal and non- seasonal level.

#Changing to stationary set

train.sta_Var4 <- diff(train3)
test.sta_Var4 <- diff(test3)

#2.1 Auto Arima model without sesonality:

set.seed(301)
model_Var4 <- auto.arima(train.sta_Var4, seasonal = FALSE)
summary(model_Var4)
## Series: train.sta_Var4 
## ARIMA(1,0,3) with zero mean 
## 
## Coefficients:
##          ar1      ma1      ma2      ma3
##       0.3077  -0.5011  -0.2231  -0.0916
## s.e.  0.1082   0.1092   0.0350   0.0490
## 
## sigma^2 estimated as 4.805:  log likelihood=-4829
## AIC=9667.99   AICc=9668.02   BIC=9696.46
## 
## Training set error measures:
##                       ME    RMSE      MAE MPE MAPE      MASE          ACF1
## Training set 0.001500922 2.19006 1.494088 NaN  Inf 0.6431976 -0.0005031034

##NOTE: AIC = 9667.99 , BIC = 9696.46 , RMSE = 2.19006

tsdisplay(residuals(model_Var4), lag.max = 45, main = '(1,1,1) Model Residuals')

#2.2 Auto Arima model with sesonality:

set.seed(301)
model2_Var4 <- auto.arima(train.sta_Var4, seasonal= TRUE)
summary(model2_Var4)
## Series: train.sta_Var4 
## ARIMA(2,0,1) with zero mean 
## 
## Coefficients:
##          ar1      ar2      ma1
##       0.6801  -0.1497  -0.8753
## s.e.  0.0273   0.0234   0.0190
## 
## sigma^2 estimated as 4.806:  log likelihood=-4829.78
## AIC=9667.55   AICc=9667.57   BIC=9690.32
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set 0.001505874 2.190842 1.493964 NaN  Inf 0.6431441 0.001093764

##NOTE: AIC =9667.55, BIC = 9690.32, RMSE = 2.190842

tsdisplay(residuals(model2_Var4), lag.max = 20, main = 'Seasonal Model Residuals')

NOTE: Above Graphs shows serious lags at 15

#2.3 Manual Arima Passing parameter order = (2,0,15) which means p(member of lags of a variable to be used as a predictor) = 2, d = 0(members of differences needed for stationarity) and q = 15 (moving average order - number of lagged forecasts error in the prediction equation)

set.seed(301)
model1_Var4 <- arima(train.sta_Var4, order = c(2,0,15))
summary(model1_Var4)
## 
## Call:
## arima(x = train.sta_Var4, order = c(2, 0, 15))
## 
## Coefficients:
##           ar1     ar2     ma1      ma2      ma3      ma4     ma5      ma6
##       -0.2732  0.2377  0.0767  -0.5772  -0.2038  -0.0118  0.0182  -0.0164
## s.e.   0.2844  0.2034  0.2843   0.2003   0.1017   0.0859  0.0451   0.0262
##           ma7      ma8     ma9    ma10    ma11    ma12     ma13     ma14
##       -0.0165  -0.0034  0.0141  0.0129  0.0165  0.0359  -0.0240  -0.0297
## s.e.   0.0257   0.0268  0.0247  0.0259  0.0258  0.0258   0.0274   0.0263
##          ma15  intercept
##       -0.0450     0.0003
## s.e.   0.0257     0.0112
## 
## sigma^2 estimated as 4.763:  log likelihood = -4821.44,  aic = 9680.88
## 
## Training set error measures:
##                        ME     RMSE      MAE MPE MAPE      MASE         ACF1
## Training set 0.0006981842 2.182491 1.489708 NaN  Inf 0.6420169 0.0001722461

##NOTE: AIC = 9680.88 , RMSE = 2.182491

tsdisplay(residuals(model1_Var4), lag.max = 20, main = 'Seasonal Model Residuals')

Note: It seems good fit.

Fit the observed with the predicted values for both models in the same plot.

#Plotting forecast for each model

par(mfrow = c(2,2))
fit_auto_train_Var4 %>% forecast(h=341) %>% autoplot() 

model1_Var4 %>% forecast(h=341) %>% autoplot() 

model2_Var4 %>% forecast(h=341) %>% autoplot() 

model_Var4 %>% forecast(h=341) %>% autoplot() 

Plot and comment on the residuals of the fitted data for both models in the same plot.

Residual Plot - to confirm no problem with this model

par(mfrow= c(2,2))
hist(fit_auto_train_Var4$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(fit_auto_train_Var4$residuals))

hist(model_Var4$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model_Var4$residuals))

hist(model1_Var4$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model1_Var4$residuals))

hist(model2_Var4$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model2_Var4$residuals))

OBSERVATION: The residuals for each model are normally distributed, which shows the good fit of the models. It is normally distributed

Testing the autocorrelation and partial autocorrelation of the residuals (using the plots and Ljung-Box Q statistic) up to an appropriate lag *

  1. ACF & PACF Plots
par(mfrow = c(4,2))
acf(fit_auto_train_Var4$residuals, main = 'Correlogram')
pacf(fit_auto_train_Var4$residuals, main = 'Partial Corelogram')

acf(model_Var4$residuals, main = 'Correlogram')
pacf(model_Var4$residuals, main = 'Partial Corelogram')

acf(model1_Var4$residuals, main = 'Correlogram')
pacf(model1_Var4$residuals, main = 'Partial Corelogram')

acf(model2_Var4$residuals, main = 'Correlogram')
pacf(model2_Var4$residuals, main = 'Partial Corelogram')

NOTE:
1. Residuals are the difference between actaual and fitted values
2. These blue lines are signficance bounds
3. This graph shows us the autocorelations for insample forecast errors
4. Do not exceed these significance bounds for lags 1 to end

#Ljunx-Box Test to check the autocorelations of the residuals

HO: No serial correlation upto 20 lags (does not exhibit lack of fit of the model)
Ha: Serial corelation is present (the model exhibits lack of fit)

We reject the null hypothesis, if p value is less than 0.05. This test is generally done for ARIMA models

Box.test(model_Var4$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model_Var4$residuals
## X-squared = 21.379, df = 20, p-value = 0.3751
Box.test(model1_Var4$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model1_Var4$residuals
## X-squared = 4.1403, df = 20, p-value = 0.9999
Box.test(model2_Var4$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model2_Var4$residuals
## X-squared = 22.829, df = 20, p-value = 0.2972

OBSERVATIONS:
For above performed Ljunx-Box Test on all ARIMA models output, we can say that p-value is more than significance level 0.05, then there is not a statistical significance.
We conclude that there is a liitle evidence of non-zero auto-corelations in the insample forecast errors at lags 1 to 20.
So we fail to rejects the null hypothesis. Hence, we conclude that there is no serial auto corelation present between lags. Hence, the model seems to be good fit.

Test and comment on the normality of the residuals.

Here, further testing the normality by plotting Normal Q-Q plots. From below plots we observed that most of the datapoints lies on the line. Hence, we could conclude that our model produces residuals that are normal

par(mfrow = c(2,2))
qqnorm(fit_auto_train_Var4$residuals)
qqnorm(model_Var4$residuals)
qqnorm(model1_Var4$residuals)
qqnorm(model2_Var4$residuals)

Validating The Model by testing the model performance with Holdout set

Here, choosing RMSE as evaluation parameter. The lower RMSE indicates a more accurate forecast.

Below, is the comparison of model. We are fitting the model to test set which is the holdout set. And checking the accuracy using RMSE(Root mean square error) evaluation parameter.

accuracy(forecast(fit_auto_train_Var4), test3) ["Test set", "RMSE"]
## [1] 4.297961
#o/p - 4.297961
accuracy(forecast(model1_Var4), test.sta_Var4) ["Test set", "RMSE"]
## [1] 2.202662
#O/P - 2.202662
accuracy(forecast(model2_Var4), test.sta_Var4) ["Test set", "RMSE"]
## [1] 2.203002
#o/p - 2.203002
accuracy(forecast(model_Var4), test.sta_Var4) ["Test set", "RMSE"]
## [1] 2.202313
#O/P - 2.202313

NOTE:- Here, the RMSE value is less of ARIMA models compared to exponential model.
The minimum RMSE is of model_Var4.

COMMENT ON BEST MODEL AND ITS PARAMETERS

We observed the best model to be ARIMA model i.e.“model_Var4.”.

summary(model_Var4)
## Series: train.sta_Var4 
## ARIMA(1,0,3) with zero mean 
## 
## Coefficients:
##          ar1      ma1      ma2      ma3
##       0.3077  -0.5011  -0.2231  -0.0916
## s.e.  0.1082   0.1092   0.0350   0.0490
## 
## sigma^2 estimated as 4.805:  log likelihood=-4829
## AIC=9667.99   AICc=9668.02   BIC=9696.46
## 
## Training set error measures:
##                       ME    RMSE      MAE MPE MAPE      MASE          ACF1
## Training set 0.001500922 2.19006 1.494088 NaN  Inf 0.6431976 -0.0005031034

Passing parameter order = (1,0,3) ,which means

p = 1,
d = 0
q = 3

For the best model, 1 member of lag is used as a predictor. As we transformed the series to stationary so members of differences needed for stationarity is 0. 3 are the number of lagged forecasts errors used in the prediction equation.
AIC = 9667.99, RMSE = 2.19006 which is minimum as compared to other models.

5. Forecasting on Relative Humidity at 850 hpa (RH85)

#Checking the attributes of time series

attributes(tsRH85)
## $tsp
## [1] 1998.00 2004.94  365.00
## 
## $class
## [1] "ts"

#Plotting the series

plot(tsRH85, main = " Relative Humidity at 850 hpa (tsRH85)", col = "blue")

From the plot, we observes that there seems to have is no trend but seems to have seasonality.

Checking the statistics and visualising the missing data in the series:

#To check the missing data

plotNA.distribution(tsRH85)

#data is missing at random 

Here, data is missing at random.

#Impute the missing values with “na_locf” function of “imputeTS package” and visualise the imputed values in the time series

tsRH85.imp <- na_locf(tsRH85)
plotNA.imputations(tsRH85, tsRH85.imp)

#Decomposition Of Time Series

decomp4 <- decompose(tsRH85.imp)
plot(decomp4, col = "red")

OBSERVATIONS:
1)There seems to have trend. 2)We observes slighty upward trend from year 2000 onwards.
3)We observe some seasonality in the series. Seasonality describes cyclical effects due to the time of the year.

Let’s explore this seasonality pattern by plotting the decomp$seasonal components such that y-axis represents Seasonality Index and x-axis has month

#Plotting the decomposed series

par(mfrow = c(1,1))
plot(decomp4$seasonal[1:365], type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")

Stationarity check of time series

We could check the stationarity of time series by ADF and KPSS Tests:

  1. Augmented Dickey- Fuller(ADF) t-statistic test for unit root
  2. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) for level or trend stationarity
adf.test(tsRH85.imp)
## Warning in adf.test(tsRH85.imp): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  tsRH85.imp
## Dickey-Fuller = -9.223, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary

NOTE: smaller p-value, hence the series is stationary

kpss.test(tsRH85.imp, null="Trend")
## Warning in kpss.test(tsRH85.imp, null = "Trend"): p-value greater than printed
## p-value
## 
##  KPSS Test for Trend Stationarity
## 
## data:  tsRH85.imp
## KPSS Trend = 0.10719, Truncation lag parameter = 8, p-value = 0.1

NOTE: p-value is greater, hence the series is stationary

acf(tsRH85.imp,lag.max = length(tsRH85.imp),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

NOTE: From the both tests results, it is confirmed that the series is stationary

#CREATING MODELS AND FORECASTING: USING EXPONENTIAL SMOOTHING AND ARIMA TECHNIQUES FOR TIME SERIES FORECASTING

#TRAIN- TEST SET SPLIT

train4 <- window(tsRH85.imp, end = c(2004, 3))
test4 <- window(tsRH85.imp, start = c(2004, 4))

#Model 1 : Auto Exponential Smoothening Model

set.seed(301)
fit_auto_train_Var5 <- forecast(train4)
summary(fit_auto_train_Var5)
## 
## Forecast method: STL +  ETS(A,N,N)
## 
## Model Information:
## ETS(A,N,N) 
## 
## Call:
##  ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend) 
## 
##   Smoothing parameters:
##     alpha = 0.0811 
## 
##   Initial states:
##     l = 0.5837 
## 
##   sigma:  0.2189
## 
##      AIC     AICc      BIC 
## 10212.74 10212.75 10229.82 
## 
## Error measures:
##                        ME      RMSE       MAE       MPE     MAPE      MASE
## Training set 0.0004792866 0.2188444 0.1751376 -61.12339 84.00862 0.6284998
##                   ACF1
## Training set 0.3314855
## 
## Forecasts:
##           Point Forecast        Lo 80     Hi 80        Lo 95     Hi 95
## 2004.0082      0.6903383  0.409749880 0.9709267  0.261215353 1.1194612
## 2004.0110      0.8526915  0.571181994 1.1342011  0.422159842 1.2832232
## 2004.0137      0.7988023  0.516374644 1.0812300  0.366866457 1.2307382
## 2004.0164      0.8775044  0.594161564 1.1608473  0.444168916 1.3108399
## 2004.0192      0.5907099  0.306454776 0.8749649  0.155979229 1.0254405
## 2004.0219      0.5955454  0.310380997 0.8807098  0.159424093 1.0316667
## 2004.0247      0.7433759  0.457305153 1.0294467  0.305868423 1.1808835
## 2004.0274      0.5316621  0.244687734 0.8186364  0.092772695 0.9705515
## 2004.0301      0.7332078  0.445332788 1.0210829  0.292940940 1.1734747
## 2004.0329      0.5895441  0.300771141 0.8783171  0.147903971 1.0311842
## 2004.0356      0.4586290  0.168960889 0.7482970  0.015619871 0.9016381
## 2004.0384      0.6023033  0.311742885 0.8928638  0.157929478 1.0466772
## 2004.0411      0.8161813  0.524731179 1.1076313  0.370446830 1.2619157
## 2004.0438      0.7291790  0.436842035 1.0215160  0.282088177 1.1762699
## 2004.0466      0.4441800  0.150958805 0.7374013 -0.004263142 0.8926232
## 2004.0493      0.6915449  0.397442088 0.9856477  0.241753460 1.1413364
## 2004.0521      0.5298240  0.234842183 0.8248057  0.078688268 0.9809596
## 2004.0548      0.3555576  0.059699523 0.6514157 -0.096918297 0.8080335
## 2004.0575      0.4860708  0.189338911 0.7828026  0.032258557 0.9398830
## 2004.0603      0.6965724  0.398969318 0.9941754  0.241427788 1.1517169
## 2004.0630      0.5864193  0.287947636 0.8848910  0.129946275 1.0428923
## 2004.0658      0.7472774  0.447939608 1.0466152  0.289479751 1.2050751
## 2004.0685      0.5206612  0.220459812 0.8208626  0.061542782 0.9797797
## 2004.0712      0.6014621  0.300399587 0.9025247  0.141026695 1.0618976
## 2004.0740      0.4397424  0.137821111 0.7416636 -0.022006343 0.9014911
## 2004.0767      0.5056265  0.202849017 0.8084040  0.042568291 0.9686847
## 2004.0795      0.7315324  0.427901046 1.0351637  0.267168326 1.1958964
## 2004.0822      0.8532957  0.548812953 1.1577785  0.387629506 1.3189620
## 2004.0849      0.7203875  0.415055656 1.0257194  0.253422740 1.1873523
## 2004.0877      0.6182861  0.312107588 0.9244647  0.150026448 1.0865459
## 2004.0904      0.6510228  0.343999838 0.9580457  0.181471711 1.1205738
## 2004.0932      0.7848989  0.477033929 1.0927639  0.314060041 1.2557378
## 2004.0959      0.5916016  0.282896815 0.9003064  0.119478381 1.0637248
## 2004.0986      0.5299656  0.220423318 0.8395078  0.056561544 1.0033696
## 2004.1014      0.4878031  0.177425579 0.7981806  0.013121663 0.9624845
## 2004.1041      0.5809627  0.269752263 0.8921732  0.105007390 1.0569181
## 2004.1068      0.6216550  0.309613797 0.9336963  0.144429145 1.0988809
## 2004.1096      0.7530308  0.440160968 1.0659006  0.274537704 1.2315238
## 2004.1123      0.7644668  0.450770642 1.0781630  0.284709925 1.2442237
## 2004.1151      0.6701589  0.355638575 0.9846793  0.189141553 1.1511763
## 2004.1178      0.7938439  0.478501494 1.1091863  0.311569309 1.2761185
## 2004.1205      0.6830619  0.366899560 0.9992242  0.199533343 1.1665904
## 2004.1233      0.4194583  0.102478209 0.7364384 -0.065320917 0.9042375
## 2004.1260      0.5918362  0.274040447 0.9096320  0.105809525 1.0778629
## 2004.1288      0.7957964  0.477186984 1.1144057  0.308525373 1.2830673
## 2004.1315      0.7070350  0.387614120 1.0264559  0.218522915 1.1955471
## 2004.1342      0.5851654  0.264935034 0.9053957  0.095415325 1.0749154
## 2004.1370      0.4927723  0.171734488 0.8138100  0.001787355 0.9837572
## 2004.1397      0.5161260  0.194282863 0.8379692  0.023909379 1.0083427
## 2004.1425      0.6292570  0.306610447 0.9519036  0.135811675 1.1227023
## 2004.1452      0.7529398  0.429491829 1.0763877  0.258268826 1.2476107
## 2004.1479      0.7023877  0.378140371 1.0266351  0.206494186 1.1982813
## 2004.1507      0.8628819  0.537837124 1.1879267  0.365768797 1.3599951
## 2004.1534      0.6068324  0.280992059 0.9326727  0.108502624 1.1051621
## 2004.1562      0.5219131  0.195279282 0.8485470  0.022369764 1.0214565
## 2004.1589      0.7264967  0.399071259 1.0539222  0.225742677 1.2272508
## 2004.1616      0.4924090  0.164193834 0.8206243 -0.009552803 0.9943709
## 2004.1644      0.4743785  0.145375452 0.8033815 -0.028788236 0.9775452
## 2004.1671      0.8039815  0.474192506 1.1337705  0.299612764 1.3083502
## 2004.1699      0.6172734  0.286700292 0.9478464  0.111705484 1.1228412
## 2004.1726      0.6384796  0.307124356 0.9698349  0.131715466 1.1452438
## 2004.1753      0.7241334  0.391997720 1.0562690  0.216175722 1.2320910
## 2004.1781      0.6788397  0.345925449 1.0117539  0.169691312 1.1879880
## 2004.1808      0.8006346  0.466943634 1.1343255  0.290298318 1.3109708
## 2004.1836      0.6075687  0.273102838 0.9420346  0.096047300 1.1190901
## 2004.1863      0.4655810  0.130341949 0.8008200 -0.047122864 0.9782848
## 2004.1890      0.3681486  0.032138198 0.7041589 -0.145734948 0.8820321
## 2004.1918      0.3647278  0.027947795 0.7015077 -0.150332749 0.8797883
## 2004.1945      0.6882615  0.350713663 1.0258093  0.172026651 1.2044963
## 2004.1973      0.7578975  0.419583639 1.0962114  0.240491080 1.2753040
## 2004.2000      0.6855802  0.346501916 1.0246584  0.167004727 1.2041556
## 2004.2027      0.7235325  0.383691545 1.0633734  0.203790636 1.2432743
## 2004.2055      0.6914548  0.350852987 1.0320567  0.170549262 1.2123604
## 2004.2082      0.6563413  0.314980170 0.9977023  0.134274527 1.1784080
## 2004.2110      0.5914371  0.249318417 0.9335557  0.068211747 1.1146624
## 2004.2137      0.6393040  0.296429444 0.9821785  0.114922635 1.1636853
## 2004.2164      0.4569210  0.113292303 0.8005498 -0.068613766 0.9824559
## 2004.2192      0.5409456  0.196564244 0.8853269  0.014259789 1.0676313
## 2004.2219      0.6058790  0.260746769 0.9510113  0.078044797 1.1337132
## 2004.2247      0.6733852  0.327503665 1.0192667  0.144405039 1.2023654
## 2004.2274      0.5141911  0.167561911 0.8608203 -0.015932512 1.0443148
## 2004.2301      0.4402605  0.092885217 0.7876358 -0.091004150 0.9715252
## 2004.2329      0.6333339  0.285214163 0.9814537  0.100930698 1.1657371
## 2004.2356      0.6125622  0.263699533 0.9614248  0.079022810 1.1461015
## 2004.2384      0.5123836  0.162779630 0.8619875 -0.022289514 1.0470566
## 2004.2411      0.7413918  0.391048083 1.0917354  0.205587348 1.2771962
## 2004.2438      0.7967543  0.445672503 1.1478362  0.259821003 1.3336877
## 2004.2466      0.9064421  0.554623635 1.2582606  0.368382187 1.4445020
## 2004.2493      0.8315019  0.478948390 1.1840555  0.292317811 1.3706861
## 2004.2521      0.5809850  0.227697936 0.9342722  0.040679035 1.1212911
## 2004.2548      0.6727809  0.318761766 1.0268001  0.131355349 1.2142065
## 2004.2575      0.6503021  0.295552432 1.0050518  0.107759297 1.1928449
## 2004.2603      0.4831357  0.127657012 0.8386144 -0.060522045 1.0267935
## 2004.2630      0.5194179  0.163211705 0.8756242 -0.025352485 1.0641884
## 2004.2658      0.5233246  0.166392287 0.8802569 -0.022556252 1.0692054
## 2004.2685      0.6681129  0.310456011 1.0257697  0.121123905 1.2151018
## 2004.2712      0.5843268  0.225946832 0.9427068  0.036231934 1.1324217
## 2004.2740      0.4994809  0.140379244 0.8585825 -0.049717675 1.0486794
## 2004.2767      0.5626058  0.202783920 0.9224276  0.012305746 1.1129058
## 2004.2795      0.5115828  0.151042197 0.8721234 -0.039816471 1.0629821
## 2004.2822      0.6927907  0.331532791 1.0540487  0.140294386 1.2452871
## 2004.2849      0.7145352  0.352561294 1.0765090  0.160943905 1.2681264
## 2004.2877      0.6984662  0.335777853 1.0611546  0.143782228 1.2531502
## 2004.2904      0.4925580  0.129156482 0.8559594 -0.063216635 1.0483325
## 2004.2932      0.5414262  0.177313033 0.9055394 -0.015436837 1.0982892
## 2004.2959      0.5121307  0.147307241 0.8769542 -0.045818647 1.0700801
## 2004.2986      0.6199850  0.254452552 0.9855174  0.060951377 1.1790186
## 2004.3014      0.7866214  0.420381434 1.1528614  0.226505698 1.3467371
## 2004.3041      0.7681789  0.401232672 1.1351250  0.206983097 1.3293746
## 2004.3068      0.6904692  0.322818131 1.0581202  0.128195436 1.2527429
## 2004.3096      0.5434475  0.175092985 0.9118020 -0.019902116 1.1067971
## 2004.3123      0.7219339  0.352877230 1.0909906  0.157510432 1.2863574
## 2004.3151      0.6435051  0.273747606 1.0132626  0.078009817 1.2090004
## 2004.3178      0.7618530  0.391395991 1.1323099  0.195287913 1.3284180
## 2004.3205      0.7427656  0.371610425 1.1139207  0.175132756 1.3103984
## 2004.3233      0.7982716  0.426419628 1.1701237  0.229573062 1.3669702
## 2004.3260      0.7701508  0.397603209 1.1426984  0.200388436 1.3399131
## 2004.3288      0.8141442  0.440902400 1.1873861  0.243320105 1.3849684
## 2004.3315      0.5940443  0.220109452 0.9679791  0.022160319 1.1659282
## 2004.3342      0.4057537  0.031127164 0.7803802 -0.167188129 0.9786955
## 2004.3370      0.6349567  0.259639780 1.0102736  0.060959002 1.2089544
## 2004.3397      0.6912215  0.315215466 1.0672276  0.116169874 1.2662732
## 2004.3425      0.7004929  0.323798939 1.0771869  0.124389200 1.2765966
## 2004.3452      0.5643836  0.187003011 0.9417642 -0.012770211 1.1415374
## 2004.3479      0.5971756  0.219109606 0.9752416  0.018973560 1.1753776
## 2004.3507      0.6025342  0.223784026 0.9812843  0.023285814 1.1817825
## 2004.3534      0.6614977  0.282064672 1.0409308  0.081204947 1.2417905
## 2004.3562      0.5065863  0.126471529 0.8867010 -0.074749059 1.0879216
## 2004.3589      0.4612126  0.080417412 0.8420078 -0.121163394 1.0435886
## 2004.3616      0.7513036  0.369829129 1.1327781  0.167888748 1.3347184
## 2004.3644      0.6670635  0.284910993 1.0492160  0.082611675 1.2515153
## 2004.3671      0.6735360  0.290706688 1.0563654  0.088049070 1.2590230
## 2004.3699      0.7291713  0.345666321 1.1126763  0.142651035 1.3156916
## 2004.3726      0.6910850  0.306905576 1.0752645  0.103533251 1.2786368
## 2004.3753      0.7352996  0.350446900 1.1201524  0.146718162 1.3238811
## 2004.3781      0.8032623  0.417737397 1.1887871  0.213652868 1.3928716
## 2004.3808      0.7303742  0.344178386 1.1165700  0.139738685 1.3210097
## 2004.3836      0.6088465  0.221980968 0.9957121  0.017186711 1.2005063
## 2004.3863      0.7944344  0.406900268 1.1819686  0.201752069 1.3871168
## 2004.3890      0.7295727  0.341371102 1.1177744  0.135869570 1.3232759
## 2004.3918      0.7368402  0.347972257 1.1257082  0.142117998 1.3315624
## 2004.3945      0.8427149  0.453181781 1.2322480  0.246975398 1.4384544
## 2004.3973      0.8203734  0.430176247 1.2105706  0.223618341 1.4171285
## 2004.4000      0.8536087  0.462748591 1.2444688  0.255839758 1.4513776
## 2004.4027      0.8503690  0.458847115 1.2418909  0.251587951 1.4491500
## 2004.4055      0.8417291  0.449546521 1.2339116  0.241937617 1.4415205
## 2004.4082      0.8615636  0.468721437 1.2544057  0.260763380 1.4623637
## 2004.4110      0.8056894  0.412188821 1.1991900  0.203882197 1.4074966
## 2004.4137      0.6987824  0.304624493 1.0929404  0.095969884 1.3015950
## 2004.4164      0.7335847  0.338770457 1.1283989  0.129768442 1.3374009
## 2004.4192      0.8435064  0.448037003 1.2389758  0.238688160 1.4483246
## 2004.4219      0.7866572  0.390533696 1.1827806  0.180838597 1.3924757
## 2004.4247      0.9348751  0.538098595 1.3316516  0.328057811 1.5416923
## 2004.4274      0.8856140  0.488185546 1.2830424  0.277799645 1.4934283
## 2004.4301      0.8839949  0.485915648 1.2820742  0.275185196 1.4928047
## 2004.4329      0.8187265  0.419997415 1.2174556  0.208922975 1.4285301
## 2004.4356      0.8088582  0.409480350 1.2082361  0.198062481 1.4196539
## 2004.4384      0.7660536  0.366028038 1.1660792  0.154267296 1.3778399
## 2004.4411      0.8003651  0.399692894 1.2010373  0.187589834 1.4131404
## 2004.4438      0.7534894  0.352171595 1.1548073  0.139726768 1.3672521
## 2004.4466      0.8263503  0.424387868 1.2283127  0.211601824 1.4410987
## 2004.4493      0.8239641  0.421358152 1.2265700  0.208231437 1.4396968
## 2004.4521      0.7992179  0.395969457 1.2024664  0.182502615 1.4159332
## 2004.4548      0.7931283  0.389238372 1.1970183  0.175431943 1.4108247
## 2004.4575      0.7532842  0.348753763 1.1578146  0.134608286 1.3719601
## 2004.4603      0.7992506  0.394080742 1.2044205  0.179596752 1.4189045
## 2004.4630      0.7382550  0.332446609 1.1440633  0.117624641 1.3588853
## 2004.4658      0.6878402  0.281394347 1.0942860  0.066234932 1.3094454
## 2004.4685      0.8700466  0.462964290 1.2771288  0.247467955 1.4926252
## 2004.4712      0.8488557  0.441137922 1.2565734  0.225305194 1.4724061
## 2004.4740      0.7790035  0.370651294 1.1873557  0.154482697 1.4035243
## 2004.4767      0.8413452  0.432359472 1.2503309  0.215855526 1.4668348
## 2004.4795      0.7740973  0.364479076 1.1837155  0.147640300 1.4005543
## 2004.4822      0.7427011  0.332451397 1.1529509  0.115278306 1.3701240
## 2004.4849      0.8324691  0.421588768 1.2433494  0.204081877 1.4608563
## 2004.4877      0.8642048  0.452694915 1.2757147  0.234854735 1.4935549
## 2004.4904      0.7045009  0.292362398 1.1166395  0.074189439 1.3348124
## 2004.4932      0.6450355  0.232269257 1.0578017  0.013764025 1.2763069
## 2004.4959      0.7268628  0.313469896 1.1402558  0.094632896 1.3590928
## 2004.4986      0.7159586  0.301939898 1.1299773  0.082771632 1.3491456
## 2004.5014      0.7945505  0.379906935 1.2091940  0.160407904 1.4286930
## 2004.5041      0.7477166  0.332449154 1.1629840  0.112619854 1.3828133
## 2004.5068      0.7089673  0.293076889 1.1248577  0.072917816 1.3450167
## 2004.5096      0.7641209  0.347608498 1.1806333  0.127120145 1.4011217
## 2004.5123      0.7028242  0.285690737 1.1199578  0.064873596 1.3407749
## 2004.5151      0.7457759  0.328022199 1.1635296  0.106876758 1.3846750
## 2004.5178      0.7400395  0.321666580 1.1584125  0.100193326 1.3798857
## 2004.5205      0.7821120  0.363120763 1.2011033  0.141320180 1.4229039
## 2004.5233      0.8162387  0.396629961 1.2358474  0.174502533 1.4579748
## 2004.5260      0.8415881  0.421362877 1.2618133  0.198909082 1.4842671
## 2004.5288      0.8292412  0.408400397 1.2500821  0.185620714 1.4728618
## 2004.5315      0.8331482  0.411692631 1.2546037  0.188587536 1.4777088
## 2004.5342      0.7926687  0.370599304 1.2147381  0.147169271 1.4381681
## 2004.5370      0.7920329  0.369350550 1.2147152  0.145596051 1.4384697
## 2004.5397      0.7814203  0.358125980 1.2047147  0.134047485 1.4287932
## 2004.5425      0.7803950  0.356489484 1.2043005  0.132087460 1.4287025
## 2004.5452      0.7513843  0.326868536 1.1759001  0.102143450 1.4006252
## 2004.5479      0.8256343  0.400509116 1.2507595  0.175461431 1.4758072
## 2004.5507      0.8094534  0.383719631 1.2351871  0.158349808 1.4605569
## 2004.5534      0.7731100  0.346768624 1.1994514  0.121077124 1.4251429
## 2004.5562      0.7743631  0.347414952 1.2013113  0.121402233 1.4273241
## 2004.5589      0.8138349  0.386280798 1.2413891  0.159947315 1.4677225
## 2004.5616      0.8141966  0.386037437 1.2423558  0.159383644 1.4690096
## 2004.5644      0.7456027  0.316839240 1.1743661  0.089865590 1.4013398
## 2004.5671      0.8361732  0.406806403 1.2655400  0.179513345 1.4928331
## 2004.5699      0.8072868  0.377317441 1.2372561  0.149705424 1.4648681
## 2004.5726      0.7594244  0.328853399 1.1899954  0.100922869 1.4179260
## 2004.5753      0.7964734  0.365301555 1.2276453  0.137052957 1.4558939
## 2004.5781      0.8088478  0.377075930 1.2406197  0.148509705 1.4691859
## 2004.5808      0.7317502  0.299379164 1.1641213  0.070495754 1.3930047
## 2004.5836      0.8617991  0.428829652 1.2947685  0.199629496 1.5239686
## 2004.5863      0.8489999  0.415432942 1.2825668  0.185916477 1.5120833
## 2004.5890      0.7503520  0.316188342 1.1845156  0.086356004 1.4143479
## 2004.5918      0.7935974  0.358837859 1.2283569  0.128690081 1.4585046
## 2004.5945      0.6962466  0.260892000 1.1316011  0.030429213 1.3620639
## 2004.5973      0.7197816  0.283832779 1.1557304  0.053055413 1.3865078
## 2004.6000      0.7477828  0.311240574 1.1843251  0.080149058 1.4154166
## 2004.6027      0.7706745  0.333539610 1.2078094  0.102134370 1.4392147
## 2004.6055      0.7626004  0.324873614 1.2003271  0.093155075 1.4320456
## 2004.6082      0.8835149  0.445197108 1.3218327  0.213165693 1.5538641
## 2004.6110      0.8246786  0.385770591 1.2635866  0.153426722 1.4959305
## 2004.6137      0.8283721  0.388874673 1.2678696  0.156218768 1.5005255
## 2004.6164      0.7917123  0.351626135 1.2317984  0.118658613 1.4647659
## 2004.6192      0.7649834  0.324309378 1.2056574  0.091030655 1.4389361
## 2004.6219      0.6603092  0.219048151 1.1015703 -0.014541357 1.3351598
## 2004.6247      0.7787516  0.336904179 1.2205990  0.103004297 1.4544988
## 2004.6274      0.8079520  0.365519094 1.2503849  0.131309251 1.4845948
## 2004.6301      0.7304504  0.287432707 1.1734681  0.052913311 1.4079875
## 2004.6329      0.8681154  0.424513729 1.3117171  0.189685188 1.5465456
## 2004.6356      0.9154196  0.471234692 1.3596045  0.236097413 1.5947418
## 2004.6384      0.8228060  0.378038698 1.2675734  0.142593087 1.5030190
## 2004.6411      0.8252123  0.379863265 1.2705613  0.144109724 1.5063149
## 2004.6438      0.7093002  0.263370269 1.1552302  0.027309200 1.3912913
## 2004.6466      0.7411020  0.294591859 1.1876122  0.058223662 1.4239804
## 2004.6493      0.8037303  0.356640699 1.2508199  0.119965773 1.4874948
## 2004.6521      0.7398815  0.292213287 1.1875498  0.055232028 1.4245311
## 2004.6548      0.8760843  0.427838128 1.3243305  0.190550932 1.5616177
## 2004.6575      0.7259528  0.277129388 1.1747761  0.039536650 1.4123689
## 2004.6603      0.7167740  0.267374236 1.1661739  0.029476347 1.4040717
## 2004.6630      0.8248458  0.374870270 1.2748213  0.136667622 1.5130239
## 2004.6658      0.9069702  0.456419740 1.3575207  0.217912722 1.5960277
## 2004.6685      0.7036291  0.252504333 1.1547538  0.013693332 1.3935648
## 2004.6712      0.7102652  0.258566999 1.1619635  0.019452402 1.4010780
## 2004.6740      0.7709246  0.318653578 1.2231956  0.079235770 1.4626134
## 2004.6767      0.7111675  0.258324462 1.1640106  0.018603827 1.4037312
## 2004.6795      0.7054254  0.252010973 1.1588398  0.011987892 1.3988628
## 2004.6822      0.8331998  0.379214789 1.2871848  0.138889645 1.5275099
## 2004.6849      0.7490313  0.294476421 1.2035862  0.053849590 1.4442131
## 2004.6877      0.6214483  0.166324166 1.0765723 -0.074603972 1.3175005
## 2004.6904      0.7269742  0.271281633 1.1826668  0.030052563 1.4238958
## 2004.6932      0.6699687  0.213708394 1.1262290 -0.027821232 1.3677587
## 2004.6959      0.5388099  0.081982562 0.9956373 -0.159847247 1.2374671
## 2004.6986      0.7514977  0.294103942 1.2088914  0.051974322 1.4510210
## 2004.7014      0.5616630  0.103703596 1.0196224 -0.138725463 1.2620514
## 2004.7041      0.6117862  0.153261837 1.0703105 -0.089466293 1.3130387
## 2004.7068      0.6306124  0.171523826 1.0897011 -0.071503006 1.3327279
## 2004.7096      0.6892057  0.229553475 1.1488578 -0.013771693 1.3921830
## 2004.7123      0.6998181  0.239603040 1.1600332 -0.004020099 1.4036563
## 2004.7151      0.6083076  0.147530358 1.0690849 -0.096390387 1.3130056
## 2004.7178      0.6170926  0.155753813 1.0784313 -0.088464176 1.3226493
## 2004.7205      0.5743151  0.112415524 1.0362147 -0.132099348 1.2807296
## 2004.7233      0.7745075  0.312047734 1.2369672  0.067236341 1.4817786
## 2004.7260      0.7957426  0.332723401 1.2587618  0.087615843 1.5038694
## 2004.7288      0.7035257  0.239947752 1.1671037 -0.005455612 1.4125071
## 2004.7315      0.5865873  0.122451165 1.0507234 -0.123247649 1.2964222
## 2004.7342      0.6275986  0.162905075 1.0922922 -0.083088835 1.3382861
## 2004.7370      0.5183230  0.053072647 0.9835733 -0.193216004 1.2298620
## 2004.7397      0.5650978  0.099291327 1.0309042 -0.147291713 1.2774873
## 2004.7425      0.7053450  0.238983098 1.1717069 -0.007893981 1.4185840
## 2004.7452      0.6431620  0.176245344 1.1100787 -0.070925424 1.3572495
## 2004.7479      0.6536181  0.186147299 1.1210890 -0.061316809 1.3685531
## 2004.7507      0.6844069  0.216382609 1.1524312 -0.031374491 1.4001883
## 2004.7534      0.5749306  0.106353492 1.0435077 -0.141696255 1.2915575
## 2004.7562      0.8002850  0.331155678 1.2694143  0.082813629 1.5177563
## 2004.7589      0.6988698  0.229188968 1.1685506 -0.019445039 1.4171846
## 2004.7616      0.7148912  0.244659479 1.1851229 -0.004266143 1.4340485
## 2004.7644      0.6205327  0.149750738 1.0913146 -0.099466159 1.3405315
## 2004.7671      0.5049602  0.033628717 0.9762917 -0.215879114 1.2257996
## 2004.7699      0.3970287 -0.074851754 0.8689092 -0.324650182 1.1187076
## 2004.7726      0.7000421  0.227613376 1.1724709 -0.022475310 1.4225596
## 2004.7753      0.7136026  0.240626188 1.1865791 -0.009752419 1.4369577
## 2004.7781      0.5398700  0.066346471 1.0133934 -0.184321722 1.2640616
## 2004.7808      0.7454226  0.271352677 1.2194925  0.020395232 1.4704499
## 2004.7836      0.5409381  0.066322411 1.0155538 -0.184923953 1.2668001
## 2004.7863      0.5875822  0.112421375 1.0627430 -0.139113576 1.3142780
## 2004.7890      0.3789150 -0.096790389 0.8546203 -0.348613597 1.1064435
## 2004.7918      0.3720228 -0.104226427 0.8482721 -0.356337561 1.1003832
## 2004.7945      0.3954460 -0.081346566 0.8722385 -0.333745299 1.1246373
## 2004.7973      0.5451753  0.067840069 1.0225105 -0.184845935 1.2751965
## 2004.8000      0.6780347  0.200157380 1.1559119 -0.052815568 1.4088849
## 2004.8027      0.6980870  0.219668331 1.1765058 -0.033591237 1.4297653
## 2004.8055      0.5978873  0.118927737 1.0768468 -0.134618127 1.3303927
## 2004.8082      0.5589633  0.079463518 1.0384630 -0.174368318 1.2922949
## 2004.8110      0.4477275 -0.032311881 0.9277668 -0.286429369 1.1818843
## 2004.8137      0.7220120  0.241433597 1.2025903 -0.012969221 1.4569931
## 2004.8164      0.5066179  0.025501091 0.9877346 -0.229186738 1.2424224
## 2004.8192      0.6161771  0.134522567 1.0978317 -0.120449953 1.3528042
## 2004.8219      0.4966312  0.014439423 0.9788229 -0.240817472 1.2340798
## 2004.8247      0.6021744  0.119446085 1.0849028 -0.136094867 1.3404437
## 2004.8274      0.5427273  0.059462945 1.0259917 -0.196361751 1.2818163
## 2004.8301      0.7179324  0.234132611 1.2017321 -0.021975513 1.4578403
## 2004.8329      0.6736324  0.189297819 1.1579670 -0.067093420 1.4143582
## 2004.8356      0.4342586 -0.050610198 0.9191274 -0.307284240 1.1758015
## 2004.8384      0.3760001 -0.109402372 0.8614025 -0.366358905 1.1183591
## 2004.8411      0.6053154  0.119379931 1.0912509 -0.137858783 1.3484896
## 2004.8438      0.6804294  0.193961468 1.1668974 -0.063559118 1.4244180
## 2004.8466      0.5143903  0.027390482 1.0013902 -0.230411668 1.2591923
## 2004.8493      0.5897788  0.102247645 1.0773100 -0.155835762 1.3353934
## 2004.8521      0.3820814 -0.105980444 0.8701433 -0.364344801 1.1285077
## 2004.8548      0.5320945  0.043502490 1.0206866 -0.215142512 1.2793316
## 2004.8575      0.5222711  0.033149498 1.0113927 -0.225775845 1.2703181
## 2004.8603      0.7177437  0.228093060 1.2073943 -0.031112321 1.4665997
## 2004.8630      0.8597176  0.369538584 1.3498967  0.110053468 1.6093818
## 2004.8658      0.6072848  0.116577917 1.0979918 -0.143186634 1.3577563
## 2004.8685      0.5255507  0.034316470 1.0167849 -0.225727215 1.2768286
## 2004.8712      0.5310275  0.039266522 1.0227884 -0.221055998 1.2831109
## 2004.8740      0.5683224  0.076035284 1.0606095 -0.184565773 1.3212106
## 2004.8767      0.8134740  0.320661296 1.3062867  0.059782001 1.5671660
## 2004.8795      0.6767441  0.183406351 1.1700819 -0.077750887 1.4312391
## 2004.8822      0.5357548  0.041892537 1.0296170 -0.219542348 1.2910519
## 2004.8849      0.6448286  0.150442393 1.1392148 -0.111269845 1.4009270
## 2004.8877      0.6667075  0.171797943 1.1616171 -0.090191354 1.4236064
## 2004.8904      0.5887509  0.093318499 1.0841833 -0.168947564 1.3464493
## 2004.8932      0.5930738  0.097119097 1.0890284 -0.165423440 1.3515710
## 2004.8959      0.6372996  0.140823221 1.1337760 -0.121995500 1.3965947
## 2004.8986      0.6209683  0.123970789 1.1179659 -0.139123827 1.3810605
## 2004.9014      0.5032821  0.005763911 1.0008003 -0.257606308 1.2641705
## 2004.9041      0.5941201  0.096081838 1.0921584 -0.167563698 1.3558039
## 2004.9068      0.5290517  0.030493873 1.0276095 -0.233426693 1.2915301
## 2004.9096      0.5314034  0.032326628 1.0304803 -0.231868681 1.2946756
## 2004.9123      0.7038493  0.204254021 1.2034446 -0.060215746 1.4679144
## 2004.9151      0.9054951  0.405381877 1.4056083  0.140637937 1.6703522
## 2004.9178      0.7163707  0.215740139 1.2170013 -0.049277691 1.4820192
## 2004.9205      0.5719845  0.070837060 1.0731320 -0.194454376 1.3384234
## 2004.9233      0.6209071  0.119243292 1.1225708 -0.146321469 1.3881356
## 2004.9260      0.5314784  0.029298857 1.0336580 -0.236538948 1.2994958
## 2004.9288      0.4856011 -0.017093722 0.9882959 -0.283204291 1.2544065
## 2004.9315      0.5517062  0.048496624 1.0549158 -0.217886429 1.3212988
## 2004.9342      0.2449924 -0.258731417 0.7487161 -0.525386677 1.0153714
## 2004.9370      0.5088668  0.004629349 1.0131043 -0.262297839 1.2800315
## 2004.9397      0.6630588  0.158308196 1.1678094 -0.108890643 1.4350083
## 2004.9425      0.6137802  0.108516887 1.1190434 -0.158953328 1.3865136
## 2004.9452      0.5375032  0.031727861 1.0432786 -0.236013455 1.3110199
## 2004.9479      0.5150377  0.008750674 1.0213247 -0.259261469 1.2893368
## 2004.9507      0.6270537  0.120255657 1.1338518 -0.148027038 1.4021345
## 2004.9534      0.5061101 -0.001198576 1.0134187 -0.269751552 1.2819717
## 2004.9562      0.5704559  0.062637210 1.0782746 -0.206185774 1.3470976
## 2004.9589      0.4453888 -0.062939450 0.9537171 -0.332032173 1.2228098
## 2004.9616      0.4662459 -0.042591379 0.9750832 -0.311953569 1.2444454
## 2004.9644      0.6087059  0.099360062 1.1180517 -0.170271326 1.3876831
## 2004.9671      0.6229421  0.113088215 1.1327959 -0.156812104 1.4026962
## 2004.9699      0.4400719 -0.070289419 0.9504333 -0.340458400 1.2206023
## 2004.9726      0.6059110  0.095042625 1.1167794 -0.175394751 1.3872167
## 2004.9753      0.6360319  0.124657059 1.1474068 -0.146048446 1.4181123
## 2004.9781      0.5470015  0.035120619 1.0588824 -0.235852750 1.3298557
## 2004.9808      0.5174789  0.005092478 1.0298653 -0.266148491 1.3011062
## 2004.9836      0.6443095  0.131418139 1.1572009 -0.140090165 1.4287092
## 2004.9863      0.5288986  0.015502732 1.0422945 -0.256272645 1.3140699
## 2004.9890      0.5947417  0.080841810 1.1086417 -0.191200377 1.3806838
## 2004.9918      0.3032528 -0.211150637 0.8176563 -0.483459373 1.0899650
## 2004.9945      0.5454123  0.030505870 1.0603188 -0.242069155 1.3328939
## 2004.9973      0.5047455 -0.010663501 1.0201545 -0.283504554 1.2929956
## 2005.0000      0.5453453  0.029434249 1.0612564 -0.243672573 1.3343632
## 2005.0027      0.7679888  0.251576168 1.2844014 -0.021796166 1.5577738
## 2005.0055      0.5411754  0.024261720 1.0580891 -0.249375867 1.3317267
## 2005.0082      0.6903383  0.172923983 1.2077526 -0.100978600 1.4816552
## 2005.0110      0.8526915  0.334777137 1.3706059  0.060609813 1.6447733
## 2005.0137      0.7988023  0.280388305 1.3172164  0.005956497 1.5916482
## 2005.0164      0.8775044  0.358591249 1.3964176  0.083895210 1.6711136
## 2005.0192      0.5907099  0.071298019 1.1101217 -0.203661996 1.3850817
## 2005.0219      0.5955454  0.075635357 1.1154554 -0.199588381 1.3906791
## 2005.0247      0.7433759  0.222968217 1.2637837 -0.052518992 1.5392709
## 2005.0274      0.5316621  0.010757116 1.0525670 -0.264993313 1.3283175
## 2005.0301      0.7332078  0.211806125 1.2546096 -0.064207272 1.5306230
## 2005.0329      0.5895441  0.067646096 1.1114421 -0.208630018 1.3877182
## 2005.0356      0.4586290 -0.063764850 0.9810228 -0.340303432 1.2575614
## 2005.0384      0.6023033  0.079414166 1.1251925 -0.197386636 1.4019933
## 2005.0411      0.8161813  0.292797215 1.3395653  0.015734443 1.6166281
## 2005.0438      0.7291790  0.205300588 1.2530575 -0.072023908 1.5303820
## 2005.0466      0.4441800 -0.080192343 0.9685524 -0.357778316 1.2461384
## 2005.0493      0.6915449  0.166679046 1.2164108 -0.111168157 1.4942580
## 2005.0521      0.5298240  0.004465076 1.0551828 -0.273643113 1.3332910
## 2005.0548      0.3555576 -0.170293799 0.8814091 -0.448662729 1.1597780
## 2005.0575      0.4860708 -0.040272753 1.0124143 -0.318902179 1.2910437
## 2005.0603      0.6965724  0.169737208 1.2234075 -0.109152472 1.5022972
## 2005.0630      0.5864193  0.059092994 1.1137456 -0.220056697 1.3928953
## 2005.0658      0.7472774  0.219460370 1.2750944 -0.059949089 1.5545039
## 2005.0685      0.5206612 -0.007646063 1.0489685 -0.287315050 1.3286375
## 2005.0712      0.6014621  0.072665050 1.1302592 -0.207263224 1.4101875
## 2005.0740      0.4397424 -0.089544090 0.9690288 -0.369731410 1.2492161
## 2005.0767      0.5056265 -0.024148831 1.0354019 -0.304594959 1.3158480
## 2005.0795      0.7315324  0.201268587 1.2617962 -0.079436110 1.5425009
## 2005.0822      0.8532957  0.322543938 1.3840475  0.041580910 1.6650106
## 2005.0849      0.7203875  0.189148159 1.2516268 -0.092072963 1.5328480
## 2005.0877      0.6182861  0.086559701 1.1500126 -0.194919278 1.4314916
## 2005.0904      0.6510228  0.118809673 1.1832359 -0.162926927 1.4649725
## 2005.0932      0.7848989  0.252199614 1.3175983 -0.029794372 1.5995922
## 2005.0959      0.5916016  0.058416496 1.1247867 -0.223834641 1.4070378
## 2005.0986      0.5299656 -0.003704841 1.0636360 -0.286212895 1.3461440
## 2005.1014      0.4878031 -0.046352238 1.0219584 -0.329116975 1.3047231
## 2005.1041      0.5809627  0.046322986 1.1156025 -0.236698202 1.3986237
## 2005.1068      0.6216550  0.086531275 1.1567788 -0.196746132 1.4400562
## 2005.1096      0.7530308  0.217423432 1.2886381 -0.066109962 1.5721715
## 2005.1123      0.7644668  0.228376340 1.3005573 -0.055412810 1.5843464
## 2005.1151      0.6701589  0.133585770 1.2067321 -0.150458906 1.4907768
## 2005.1178      0.7938439  0.256788467 1.3308993 -0.027511505 1.6151993
## 2005.1205      0.6830619  0.145524606 1.2205991 -0.139030433 1.5051542
## 2005.1233      0.4194583 -0.118560362 0.9574770 -0.403370239 1.2422868
## 2005.1260      0.5918362  0.053336585 1.1303359 -0.231727903 1.4154003
## 2005.1288      0.7957964  0.256816173 1.3347765 -0.028502699 1.6200954
## 2005.1315      0.7070350  0.167574715 1.2464953 -0.117998314 1.5320683
## 2005.1342      0.5851654  0.045225406 1.1251054 -0.240601555 1.4109323
## 2005.1370      0.4927723 -0.047646979 1.0331915 -0.333727645 1.3192722
## 2005.1397      0.5161260 -0.024772044 1.0570241 -0.311106191 1.3433583
## 2005.1425      0.6292570  0.087880514 1.1706335 -0.198706890 1.4572209
## 2005.1452      0.7529398  0.211085296 1.2947943 -0.075755141 1.5816347
## 2005.1479      0.7023877  0.160055679 1.2447198 -0.127037567 1.5318130
## 2005.1507      0.8628819  0.320072727 1.4056911  0.032726893 1.6930370
## 2005.1534      0.6068324  0.063546425 1.1501183 -0.224051775 1.4377165
## 2005.1562      0.5219131 -0.021849108 1.0656754 -0.309699453 1.3535257
## 2005.1589      0.7264967  0.182258606 1.2707349 -0.105843663 1.5588372
## 2005.1616      0.4924090 -0.052304576 1.0371227 -0.340658549 1.3254766
## 2005.1644      0.4743785 -0.070810196 1.0195672 -0.359415653 1.3081726
## 2005.1671      0.8039815  0.258318151 1.3496448 -0.030538571 1.6385015
## 2005.1699      0.6172734  0.071135775 1.1634109 -0.217971994 1.4525187
## 2005.1726      0.6384796  0.091868234 1.1850910 -0.197490364 1.4744496
## 2005.1753      0.7241334  0.177048560 1.2712182 -0.112560650 1.5608274
## 2005.1781      0.6788397  0.131281831 1.2263975 -0.158577775 1.5162571
## 2005.1808      0.8006346  0.252604149 1.3486650 -0.037505635 1.6387748
## 2005.1836      0.6075687  0.059066091 1.1560713 -0.231293657 1.4464311
## 2005.1863      0.4655810 -0.083393446 1.0145554 -0.374002943 1.3051649
## 2005.1890      0.3681486 -0.181297219 0.9175944 -0.472156249 1.2084534
## 2005.1918      0.3647278 -0.185189006 0.9146445 -0.476297357 1.2057529
## 2005.1945      0.6882615  0.137874126 1.2386488 -0.153483331 1.5300063
## 2005.1973      0.7578975  0.207040025 1.3087550 -0.084566326 1.6003614
## 2005.2000      0.6855802  0.134252895 1.2369075 -0.157602138 1.5287625
## 2005.2027      0.7235325  0.171735799 1.2753291 -0.120367704 1.5674326
## 2005.2055      0.6914548  0.139189207 1.2437205 -0.153162556 1.5360722
## 2005.2082      0.6563413  0.103607057 1.2090755 -0.188992754 1.5016753
## 2005.2110      0.5914371  0.038234682 1.1446394 -0.254612967 1.4374871
## 2005.2137      0.6393040  0.085633811 1.1929741 -0.207461466 1.4860694
## 2005.2164      0.4569210 -0.097216496 1.0110586 -0.390559193 1.3044013
## 2005.2192      0.5409456 -0.013658980 1.0955501 -0.307248888 1.3891400
## 2005.2219      0.6058790  0.050807872 1.1609501 -0.243029039 1.4547871
## 2005.2247      0.6733852  0.117847857 1.2289225 -0.176235850 1.5230063
## 2005.2274      0.5141911 -0.041812037 1.0701943 -0.336142332 1.3645246
## 2005.2301      0.4402605 -0.116208089 0.9967291 -0.410784767 1.2913058
## 2005.2329      0.6333339  0.076400287 1.1902675 -0.218422567 1.4850904
## 2005.2356      0.6125622  0.055163887 1.1699604 -0.239904938 1.4650293
## 2005.2384      0.5123836 -0.045478977 1.0702461 -0.340793568 1.3655607
## 2005.2411      0.7413918  0.183065332 1.2997182 -0.112494820 1.5952783
## 2005.2438      0.7967543  0.237964435 1.3555443 -0.057841075 1.6513498
## 2005.2466      0.9064421  0.347189084 1.4656951  0.051138419 1.7617458
## 2005.2493      0.8315019  0.271786201 1.3912177 -0.024509415 1.6875133
## 2005.2521      0.5809850  0.020806962 1.1411631 -0.275733404 1.4377035
## 2005.2548      0.6727809  0.112140868 1.2334210 -0.184644045 1.5302059
## 2005.2575      0.6503021  0.089200480 1.2114037 -0.207828780 1.5084330
## 2005.2603      0.4831357 -0.078427116 1.0446985 -0.375700521 1.3419719
## 2005.2630      0.5194179 -0.042605712 1.0814416 -0.340123062 1.3789589
## 2005.2658      0.5233246 -0.039159526 1.0858087 -0.336920621 1.3835698
## 2005.2685      0.6681129  0.105168706 1.2310570 -0.192835935 1.5290617
## 2005.2712      0.5843268  0.020922945 1.1477307 -0.277325043 1.4459787
## 2005.2740      0.4994809 -0.064382305 1.0633441 -0.362873441 1.3618352
## 2005.2767      0.5626058 -0.001716366 1.1269279 -0.300450453 1.4256620
## 2005.2795      0.5115828 -0.053197891 1.0763635 -0.352174731 1.3753403
## 2005.2822      0.6927907  0.127551843 1.2580296 -0.171667553 1.5572490
## 2005.2849      0.7145352  0.148838435 1.2802319 -0.150623321 1.5796936
## 2005.2877      0.6984662  0.132312040 1.2646204 -0.167391879 1.5643243
## 2005.2904      0.4925580 -0.074053320 1.0591692 -0.373999208 1.3591151
## 2005.2932      0.5414262 -0.025641786 1.1084942 -0.325829448 1.4086819
## 2005.2959      0.5121307 -0.055393617 1.0796551 -0.355822856 1.3800843
## 2005.2986      0.6199850  0.052004643 1.1879653 -0.248665982 1.4886359
## 2005.3014      0.7866214  0.218185467 1.3550574 -0.082726348 1.6559692
## 2005.3041      0.7681789  0.199287647 1.3370701 -0.101865166 1.6382229
## 2005.3068      0.6904692  0.121123056 1.2598153 -0.180270562 1.5612089
## 2005.3096      0.5434475 -0.026353125 1.1132481 -0.327987356 1.4148824
## 2005.3123      0.7219339  0.151679105 1.2921887 -0.150195547 1.5940633
## 2005.3151      0.6435051  0.072796494 1.2142137 -0.229318387 1.5163286
## 2005.3178      0.7618530  0.190690928 1.3330150 -0.111663992 1.6353699
## 2005.3205      0.7427656  0.171150452 1.3143807 -0.131444317 1.6169755
## 2005.3233      0.7982716  0.226203793 1.3703395 -0.076630634 1.6731739
## 2005.3260      0.7701508  0.197630566 1.3426710 -0.105443330 1.6457449
## 2005.3288      0.8141442  0.241172011 1.3871165 -0.062141166 1.6904296
## 2005.3315      0.5940443  0.020620384 1.1674681 -0.282931884 1.4710204
## 2005.3342      0.4057537 -0.168121511 0.9796289 -0.471912681 1.2834200
## 2005.3370      0.6349567  0.060630580 1.2092828 -0.243399306 1.5133127
## 2005.3397      0.6912215  0.116444826 1.2659983 -0.187823588 1.5702667
## 2005.3425      0.7004929  0.125265951 1.2757199 -0.179240804 1.5802266
## 2005.3452      0.5643836 -0.011293226 1.1400605 -0.316038136 1.4448054
## 2005.3479      0.5971756  0.021049223 1.1733020 -0.283933656 1.4782848
## 2005.3507      0.6025342  0.025958608 1.1791097 -0.279262054 1.4843304
## 2005.3534      0.6614977  0.084473335 1.2385221 -0.220984925 1.5439804
## 2005.3562      0.5065863 -0.070886604 1.0840591 -0.376582278 1.3897548
## 2005.3589      0.4612126 -0.116708390 1.0391336 -0.422641293 1.3450665
## 2005.3616      0.7513036  0.172934791 1.3296724 -0.133235158 1.6358423
## 2005.3644      0.6670635  0.088247257 1.2458797 -0.218159553 1.5522866
## 2005.3671      0.6735360  0.094272701 1.2527994 -0.212370789 1.5594429
## 2005.3699      0.7291713  0.149461232 1.3088814 -0.157418754 1.6157614
## 2005.3726      0.6910850  0.110928541 1.2712415 -0.196187760 1.5783578
## 2005.3753      0.7352996  0.154697080 1.3159022 -0.152655353 1.6232546
## 2005.3781      0.8032623  0.222213959 1.3843105 -0.085374426 1.6918989
## 2005.3808      0.7303742  0.148880502 1.3118678 -0.158943654 1.6196920
## 2005.3836      0.6088465  0.026907814 1.1907852 -0.281151932 1.4988450
## 2005.3863      0.7944344  0.212051029 1.3768179 -0.096244127 1.6851130
## 2005.3890      0.7295727  0.146744964 1.3124005 -0.161785423 1.6209309
## 2005.3918      0.7368402  0.153568413 1.3201120 -0.155197025 1.6288774
## 2005.3945      0.8427149  0.258999429 1.4264304 -0.050000882 1.7354307
## 2005.3973      0.8203734  0.236214590 1.4045322 -0.073020415 1.7137673
## 2005.4000      0.8536087  0.269006837 1.4382105 -0.040462684 1.7476800
## 2005.4027      0.8503690  0.265324478 1.4354135 -0.044379381 1.7451174
## 2005.4055      0.8417291  0.256242219 1.4272159 -0.053695802 1.7371540
## 2005.4082      0.8615636  0.275634692 1.4474924 -0.034537314 1.7576644
## 2005.4110      0.8056894  0.219318861 1.3920599 -0.091086953 1.7024658
## 2005.4137      0.6987824  0.111970550 1.2855943 -0.198668896 1.5962338
## 2005.4164      0.7335847  0.146331768 1.3208376 -0.164541135 1.6317105
## 2005.4192      0.8435064  0.255812811 1.4312000 -0.055293373 1.7423061
## 2005.4219      0.7866572  0.198523247 1.3747911 -0.112816044 1.6861304
## 2005.4247      0.9348751  0.346301140 1.5234490  0.034728916 1.8350212
## 2005.4274      0.8856140  0.296600340 1.4746276 -0.015204642 1.7864326
## 2005.4301      0.8839949  0.294541952 1.4734479 -0.017495614 1.7854855
## 2005.4329      0.8187265  0.228834495 1.4086186 -0.083435483 1.7208885
## 2005.4356      0.8088582  0.218527474 1.3991890 -0.093974742 1.7116912
## 2005.4384      0.7660536  0.175284480 1.3568227 -0.137449803 1.6695570
## 2005.4411      0.8003651  0.209157931 1.3915723 -0.103808245 1.7045385
## 2005.4438      0.7534894  0.161844510 1.3451343 -0.151353389 1.6583322
## 2005.4466      0.8263503  0.234267949 1.4184326 -0.079161501 1.7318620
## 2005.4493      0.8239641  0.231444688 1.4164835 -0.082216142 1.7301443
## 2005.4521      0.7992179  0.206261745 1.3921741 -0.107630295 1.7060661
## 2005.4548      0.7931283  0.199735710 1.3865210 -0.114387369 1.7006440
## 2005.4575      0.7532842  0.159455454 1.3471129 -0.154898495 1.6614669
## 2005.4603      0.7992506  0.204986094 1.3935152 -0.109598554 1.7080998
## 2005.4630      0.7382550  0.143554935 1.3329550 -0.171260245 1.6477702
## 2005.4658      0.6878402  0.092704962 1.2829754 -0.222340580 1.5980209
## 2005.4685      0.8700466  0.274476513 1.4656166 -0.040799223 1.7808923
## 2005.4712      0.8488557  0.252851078 1.4448602 -0.062654684 1.7603660
## 2005.4740      0.7790035  0.182564711 1.3754423 -0.133170910 1.6911779
## 2005.4767      0.8413452  0.244472480 1.4382178 -0.071492832 1.7541832
## 2005.4795      0.7740973  0.176791012 1.3714035 -0.139403825 1.6875984
## 2005.4822      0.7427011  0.144961599 1.3404407 -0.171462596 1.6568649
## 2005.4849      0.8324691  0.234296580 1.4306416 -0.082356807 1.7472949
## 2005.4877      0.8642048  0.265599683 1.4628099 -0.051282730 1.7796923
## 2005.4904      0.7045009  0.105463474 1.3035384 -0.211647800 1.6206497
## 2005.4932      0.6450355  0.045565994 1.2445049 -0.271773975 1.5618449
## 2005.4959      0.7268628  0.126961652 1.3267640 -0.190606848 1.6443325
## 2005.4986      0.7159586  0.115626036 1.3162912 -0.202170830 1.6340880
## 2005.5014      0.7945505  0.193786820 1.3953141 -0.124238249 1.7133392
## 2005.5041      0.7477166  0.146522155 1.3489110 -0.171730953 1.6671641
## 2005.5068      0.7089673  0.107342378 1.3105922 -0.211138606 1.6290732
## 2005.5096      0.7641209  0.162065851 1.3661760 -0.156642845 1.6848847
## 2005.5123      0.7028242  0.100339334 1.3053092 -0.218596912 1.6242454
## 2005.5151      0.7457759  0.142861423 1.3486903 -0.176302211 1.6678540
## 2005.5178      0.7400395  0.136695818 1.3433832 -0.182695043 1.6627741
## 2005.5205      0.7821120  0.178339404 1.3858847 -0.141278521 1.7055026
## 2005.5233      0.8162387  0.212037400 1.4204399 -0.107807429 1.7402848
## 2005.5260      0.8415881  0.236958509 1.4462177 -0.083113062 1.7662893
## 2005.5288      0.8292412  0.224183622 1.4342988 -0.096114530 1.7545970
## 2005.5315      0.8331482  0.227662853 1.4386335 -0.092861722 1.7591581
## 2005.5342      0.7926687  0.186755929 1.3985814 -0.133994907 1.7193323
## 2005.5370      0.7920329  0.185692989 1.3983727 -0.135283950 1.7193497
## 2005.5397      0.7814203  0.174653646 1.3881870 -0.146549237 1.7093899
## 2005.5425      0.7803950  0.173201792 1.3875882 -0.148226875 1.7090169
## 2005.5452      0.7513843  0.143764907 1.3590038 -0.177889386 1.6806580
## 2005.5479      0.8256343  0.217588972 1.4336797 -0.104290789 1.7555594
## 2005.5507      0.8094534  0.200982397 1.4179243 -0.121122674 1.7400294
## 2005.5534      0.7731100  0.164213730 1.3820063 -0.158116493 1.7043365
## 2005.5562      0.7743631  0.165041830 1.3836845 -0.157513388 1.7062397
## 2005.5589      0.8138349  0.204088883 1.4235810 -0.118691174 1.7463610
## 2005.5616      0.8141966  0.204026167 1.4243671 -0.118978572 1.7473719
## 2005.5644      0.7456027  0.135008056 1.3561973 -0.188221209 1.6794266
## 2005.5671      0.8361732  0.225154749 1.4471917 -0.098298885 1.7706453
## 2005.5699      0.8072868  0.195844765 1.4187288 -0.127833084 1.7424066
## 2005.5726      0.7594244  0.147559150 1.3712897 -0.176342758 1.6951916
## 2005.5753      0.7964734  0.184185187 1.4087617 -0.139940626 1.7328875
## 2005.5781      0.8088478  0.196136897 1.4215587 -0.128212665 1.7459083
## 2005.5808      0.7317502  0.118616926 1.3448835 -0.205956232 1.6694567
## 2005.5836      0.8617991  0.248243671 1.4753544 -0.076552928 1.8001510
## 2005.5863      0.8489999  0.235022682 1.4629770 -0.089997206 1.7879969
## 2005.5890      0.7503520  0.135953271 1.3647507 -0.189289751 1.6899937
## 2005.5918      0.7935974  0.178777446 1.4084173 -0.146688558 1.7338833
## 2005.5945      0.6962466  0.081005718 1.3114874 -0.244683115 1.6371763
## 2005.5973      0.7197816  0.104120103 1.3354431 -0.221791407 1.6613546
## 2005.6000      0.7477828  0.131700983 1.3638647 -0.194433051 1.6899987
## 2005.6027      0.7706745  0.154172584 1.3871764 -0.172183823 1.7135328
## 2005.6055      0.7626004  0.145678638 1.3795221 -0.180899991 1.7061007
## 2005.6082      0.8835149  0.266173668 1.5008561 -0.060627032 1.8276568
## 2005.6110      0.8246786  0.206918176 1.4424390 -0.120104443 1.7694617
## 2005.6137      0.8283721  0.210192774 1.4465515 -0.117051614 1.7737959
## 2005.6164      0.7917123  0.173114247 1.4103103 -0.154351760 1.7377763
## 2005.6192      0.7649834  0.145966997 1.3839998 -0.181720479 1.7116872
## 2005.6219      0.6603092  0.040874777 1.2797437 -0.287034018 1.6076525
## 2005.6247      0.7787516  0.158899313 1.3986038 -0.169230652 1.7267338
## 2005.6274      0.8079520  0.187682242 1.4282218 -0.140668745 1.7565728
## 2005.6301      0.7304504  0.109763375 1.3511374 -0.218808484 1.6797093
## 2005.6329      0.8681154  0.247011426 1.4892194 -0.081781158 1.8180120
## 2005.6356      0.9154196  0.293898930 1.5369402 -0.035114229 1.8659534
## 2005.6384      0.8228060  0.200868992 1.4447431 -0.128364596 1.7739767
## 2005.6411      0.8252123  0.202859132 1.4475655 -0.126594736 1.7770193
## 2005.6438      0.7093002  0.086531228 1.3320693 -0.243142774 1.6617433
## 2005.6466      0.7411020  0.117917432 1.3642866 -0.211976558 1.6941806
## 2005.6493      0.8037303  0.180130409 1.4273301 -0.149983420 1.7574440
## 2005.6521      0.7398815  0.115866661 1.3638964 -0.214466862 1.6942299
## 2005.6548      0.8760843  0.251654695 1.5005139 -0.078898376 1.8310670
## 2005.6575      0.7259528  0.101108679 1.3507968 -0.229663794 1.6815693
## 2005.6603      0.7167740  0.091515784 1.3420323 -0.239475946 1.6730240
## 2005.6630      0.8248458  0.199173612 1.4505180 -0.132037229 1.7817288
## 2005.6658      0.9069702  0.280884413 1.5330560 -0.050545395 1.8644858
## 2005.6685      0.7036291  0.077129877 1.3301282 -0.254518753 1.6617769
## 2005.6712      0.7102652  0.083352957 1.3371775 -0.248514351 1.6690448
## 2005.6740      0.7709246  0.143599494 1.3982497 -0.188486347 1.7303355
## 2005.6767      0.7111675  0.083429885 1.3389052 -0.248874347 1.6712094
## 2005.6795      0.7054254  0.077275450 1.3335753 -0.255247029 1.6660978
## 2005.6822      0.8331998  0.204637873 1.4617617 -0.128102709 1.7945023
## 2005.6849      0.7490313  0.120057664 1.3780050 -0.212900878 1.7109635
## 2005.6877      0.6214483 -0.007936875 1.2508334 -0.341113235 1.5840097
## 2005.6904      0.7269742  0.097177866 1.3567705 -0.236216171 1.6901646
## 2005.6932      0.6699687  0.039761461 1.3001760 -0.293850110 1.6337875
## 2005.6959      0.5388099 -0.091807976 1.1694279 -0.425636938 1.5032568
## 2005.6986      0.7514977  0.120469364 1.3825260 -0.213576849 1.7165722
## 2005.7014      0.5616630 -0.069775455 1.1931014 -0.404038777 1.5273648
## 2005.7041      0.6117862 -0.020062120 1.2436345 -0.354542411 1.5781148
## 2005.7068      0.6306124 -0.001645466 1.2628703 -0.336342584 1.5975675
## 2005.7096      0.6892057  0.056538421 1.3218729 -0.278375385 1.6567867
## 2005.7123      0.6998181  0.066741798 1.3328944 -0.268388555 1.6680248
## 2005.7151      0.6083076 -0.025177495 1.2417927 -0.360524255 1.5771395
## 2005.7178      0.6170926 -0.016801074 1.2509862 -0.352364102 1.5865493
## 2005.7205      0.5743151 -0.059986815 1.2086170 -0.395765971 1.5443962
## 2005.7233      0.7745075  0.139797526 1.4092174 -0.196197620 1.7452126
## 2005.7260      0.7957426  0.160624907 1.4308603 -0.175586089 1.7670713
## 2005.7288      0.7035257  0.068000559 1.3390509 -0.268426150 1.6754776
## 2005.7315      0.5865873 -0.049345138 1.2225197 -0.385987421 1.5591620
## 2005.7342      0.6275986 -0.008740748 1.2639380 -0.345598468 1.6007957
## 2005.7370      0.5183230 -0.118423104 1.1550691 -0.455496122 1.4921421
## 2005.7397      0.5650978 -0.072054758 1.2022503 -0.409342936 1.5395385
## 2005.7425      0.7053450  0.067786275 1.3429037 -0.269716927 1.6804069
## 2005.7452      0.6431620  0.005197381 1.2811267 -0.332520709 1.6188448
## 2005.7479      0.6536181  0.015247796 1.2919885 -0.322685044 1.6299213
## 2005.7507      0.6844069  0.045631167 1.3231827 -0.292516286 1.6613301
## 2005.7534      0.5749306 -0.064250285 1.2141115 -0.402612216 1.5524735
## 2005.7562      0.8002850  0.160699170 1.4398708 -0.177877103 1.7784471
## 2005.7589      0.6988698  0.058879337 1.3388602 -0.279911143 1.6776507
## 2005.7616      0.7148912  0.074496334 1.3552860 -0.264508216 1.6942906
## 2005.7644      0.6205327 -0.020266311 1.2613316 -0.359484798 1.6005501
## 2005.7671      0.5049602 -0.136242623 1.1461631 -0.475674911 1.4855954
## 2005.7699      0.3970287 -0.244577772 1.0386352 -0.584223726 1.3782811
## 2005.7726      0.7000421  0.058032297 1.3420520 -0.281827189 1.6819115
## 2005.7753      0.7136026  0.071189665 1.3560156 -0.268883219 1.6960885
## 2005.7781      0.5398700 -0.102945876 1.1826858 -0.443232024 1.5229719
## 2005.7808      0.7454226  0.102204126 1.3886410 -0.238295152 1.7291403
## 2005.7836      0.5409381 -0.102682720 1.1845589 -0.443394996 1.5252712
## 2005.7863      0.5875822 -0.056440712 1.2316051 -0.397365852 1.5725303
## 2005.7890      0.3789150 -0.265509806 1.0233398 -0.606647678 1.3644776
## 2005.7918      0.3720228 -0.272803545 1.0168492 -0.614154016 1.3581997
## 2005.7945      0.3954460 -0.249781757 1.0406737 -0.591344694 1.3822367
## 2005.7973      0.5451753 -0.100453564 1.1908041 -0.442228835 1.5325794
## 2005.8000      0.6780347  0.032004939 1.3240644 -0.309982535 1.6660518
## 2005.8027      0.6980870  0.051656716 1.3445174 -0.290542829 1.6867169
## 2005.8055      0.5978873 -0.048943416 1.2447180 -0.391354901 1.5871295
## 2005.8082      0.5589633 -0.088267535 1.2061941 -0.430890828 1.5488174
## 2005.8110      0.4477275 -0.199903196 1.0953582 -0.542738167 1.4381931
## 2005.8137      0.7220120  0.073981662 1.3700423 -0.269064856 1.7130888
## 2005.8164      0.5066179 -0.141811822 1.1550475 -0.485069757 1.4983055
## 2005.8192      0.6161771 -0.032651680 1.2650059 -0.376120901 1.6084751
## 2005.8219      0.4966312 -0.152596513 1.1458589 -0.496276890 1.4895392
## 2005.8247      0.6021744 -0.047451891 1.2518008 -0.391343296 1.5956922
## 2005.8274      0.5427273 -0.107297424 1.1927520 -0.451399727 1.5368543
## 2005.8301      0.7179324  0.067509499 1.3683552 -0.276803571 1.7126683
## 2005.8329      0.6736324  0.022811617 1.3244532 -0.321712093 1.6689769
## 2005.8356      0.4342586 -0.216959837 1.0854771 -0.561694058 1.4302113
## 2005.8384      0.3760001 -0.275615794 1.0276159 -0.620560397 1.3725605
## 2005.8411      0.6053154 -0.046697617 1.2573285 -0.391852474 1.6024833
## 2005.8438      0.6804294  0.028019451 1.3328394 -0.317345533 1.6782044
## 2005.8466      0.5143903 -0.138416345 1.1671970 -0.483991327 1.5127720
## 2005.8493      0.5897788 -0.063424331 1.2429819 -0.409209183 1.5887668
## 2005.8521      0.3820814 -0.271517907 1.0356808 -0.617512503 1.3816754
## 2005.8548      0.5320945 -0.121900796 1.1860899 -0.468105008 1.5322941
## 2005.8575      0.5222711 -0.132119946 1.1766622 -0.478533648 1.5230759
## 2005.8603      0.7177437  0.062957124 1.3725302 -0.283665941 1.7191533
## 2005.8630      0.8597176  0.204535824 1.5148995 -0.142296477 1.8617318
## 2005.8658      0.6072848 -0.048291998 1.2628617 -0.395333410 1.6099031
## 2005.8685      0.5255507 -0.130420929 1.1815223 -0.477671325 1.5287727
## 2005.8712      0.5310275 -0.125338689 1.1873936 -0.472797945 1.5348529
## 2005.8740      0.5683224 -0.088438066 1.2250829 -0.436106054 1.5727508
## 2005.8767      0.8134740  0.156319482 1.4706285 -0.191557114 1.8185051
## 2005.8795      0.6767441  0.019195750 1.3342925 -0.328889330 1.6823776
## 2005.8822      0.5357548 -0.122187174 1.1936968 -0.470480612 1.5419902
## 2005.8849      0.6448286 -0.013506750 1.3031639 -0.362008421 1.6516656
## 2005.8877      0.6667075  0.007979049 1.3254360 -0.340730732 1.6741457
## 2005.8904      0.5887509 -0.070370464 1.2478722 -0.419288230 1.5967900
## 2005.8932      0.5930738 -0.066440252 1.2525878 -0.415565880 1.6017134
## 2005.8959      0.6372996 -0.022606831 1.2972060 -0.371940196 1.6465394
## 2005.8986      0.6209683 -0.039330280 1.2812670 -0.388871259 1.6308080
## 2005.9014      0.5032821 -0.157408487 1.1639727 -0.507156957 1.5137212
## 2005.9041      0.5941201 -0.066962202 1.2552024 -0.416918040 1.6051583
## 2005.9068      0.5290517 -0.132422120 1.1905255 -0.482585203 1.5406886
## 2005.9096      0.5314034 -0.130461626 1.1932685 -0.480831832 1.5436387
## 2005.9123      0.7038493  0.041593197 1.3661054 -0.308984009 1.7166826
## 2005.9151      0.9054951  0.242848177 1.5681420 -0.107935907 1.9189261
## 2005.9178      0.7163707  0.053333257 1.3794082 -0.297657583 1.7303991
## 2005.9205      0.5719845 -0.091443308 1.2354123 -0.442640782 1.5866098
## 2005.9233      0.6209071 -0.042910865 1.2847250 -0.394314852 1.6361290
## 2005.9260      0.5314784 -0.132729391 1.1956862 -0.484339769 1.5472966
## 2005.9288      0.4856011 -0.178996361 1.1501986 -0.530813010 1.5020152
## 2005.9315      0.5517062 -0.113280705 1.2166931 -0.465303504 1.5687159
## 2005.9342      0.2449924 -0.420383736 0.9103685 -0.772612563 1.2625973
## 2005.9370      0.5088668 -0.156898255 1.1746319 -0.509332991 1.5270666
## 2005.9397      0.6630588 -0.003094989 1.3292126 -0.355735513 1.6818532
## 2005.9425      0.6137802 -0.052762173 1.2803225 -0.405608366 1.6331687
## 2005.9452      0.5375032 -0.129427369 1.2044339 -0.482479110 1.5574856
## 2005.9479      0.5150377 -0.152281017 1.1823563 -0.505538186 1.5356135
## 2005.9507      0.6270537 -0.040652785 1.2947602 -0.394115264 1.6482227
## 2005.9534      0.5061101 -0.161984060 1.1742042 -0.515651729 1.5278719
## 2005.9562      0.5704559 -0.098025603 1.2389374 -0.451898344 1.5928102
## 2005.9589      0.4453888 -0.223479882 1.1142575 -0.577557575 1.4683352
## 2005.9616      0.4662459 -0.203009714 1.1355015 -0.557292242 1.4897841
## 2005.9644      0.6087059 -0.060936462 1.2783482 -0.415423705 1.6328355
## 2005.9671      0.6229421 -0.047086782 1.2929709 -0.401778623 1.6476627
## 2005.9699      0.4400719 -0.230343172 1.1104870 -0.585239492 1.4653834
## 2005.9726      0.6059110 -0.064890166 1.2767121 -0.419990847 1.6318128
## 2005.9753      0.6360319 -0.035155050 1.3072189 -0.390459976 1.6625238
## 2005.9781      0.5470015 -0.124571088 1.2185741 -0.480080140 1.5740831
## 2005.9808      0.5174789 -0.154479105 1.1894368 -0.510192168 1.5451499
## 2005.9836      0.6443095 -0.028033598 1.3166527 -0.383950553 1.6725696
## 2005.9863      0.5288986 -0.143829435 1.2016267 -0.499950166 1.5577474
## 2005.9890      0.5947417 -0.078371062 1.2678545 -0.434695452 1.6241789
## 2005.9918      0.3032528 -0.370244488 0.9767501 -0.726772422 1.3332780
## 2005.9945      0.5454123 -0.128469233 1.2192939 -0.485200594 1.5760253
## 2005.9973      0.5047455 -0.169520128 1.1790112 -0.526454800 1.5359458
## 2006.0000      0.5453453 -0.129304172 1.2199948 -0.486442040 1.5771327
## 2006.0027      0.7679888  0.092955681 1.4430219 -0.264385266 1.8003629
## 2006.0055      0.5411754 -0.134241100 1.2165920 -0.491785011 1.5741359

##NOTE: AIC = 10212.74, BIC = 10229.82 , RMSE = 0.2188444 Also, the headline of the plot below contains ETS(A,N,N) showing that the automated model characterized error of the “tsWSR_PK” SERIES as Additative (A), trend as None (A) and seasonality as None (N) which is equivalent to simple exponential model(ses) with only alpha parameter i.e level.

#Model 2 : ARIMA Model

#2.1 Auto Arima model without sesonality:

set.seed(301)
model_Var5 <- auto.arima(train4, seasonal = FALSE)
summary(model_Var5)
## Series: train4 
## ARIMA(3,0,1) with non-zero mean 
## 
## Coefficients:
##          ar1      ar2     ar3      ma1    mean
##       1.3782  -0.5089  0.1162  -0.9370  0.5683
## s.e.  0.0253   0.0348  0.0218   0.0145  0.0209
## 
## sigma^2 estimated as 0.05254:  log likelihood=121.04
## AIC=-230.07   AICc=-230.03   BIC=-195.91
## 
## Training set error measures:
##                        ME      RMSE       MAE       MPE     MAPE      MASE
## Training set 0.0003031604 0.2289515 0.1832716 -71.02974 94.35383 0.6576897
##                      ACF1
## Training set 0.0003856439

##NOTE: AIC = -230.07 , BIC = -195.91 , RMSE = 0.2289515

tsdisplay(residuals(model_Var5), lag.max = 45, main = '(1,1,1) Model Residuals')

#2.2 Auto Arima model with sesonality:

set.seed(301)
model2_Var5 <- auto.arima(train4, seasonal= TRUE)
summary(model2_Var5)
## Series: train4 
## ARIMA(5,0,2) with non-zero mean 
## 
## Coefficients:
##          ar1     ar2      ar3     ar4     ar5     ma1      ma2    mean
##       0.4184  0.8105  -0.3610  0.0955  0.0078  0.0255  -0.9012  0.5693
## s.e.  0.0330  0.0375   0.0305  0.0243  0.0227  0.0253   0.0245  0.0208
## 
## sigma^2 estimated as 0.0525:  log likelihood=123.35
## AIC=-228.69   AICc=-228.61   BIC=-177.46
## 
## Training set error measures:
##                        ME      RMSE       MAE       MPE     MAPE      MASE
## Training set 7.119779e-05 0.2287086 0.1830377 -71.09868 94.38878 0.6568503
##                       ACF1
## Training set -0.0001554728

##NOTE: AIC = -228.69 , BIC = -177.46 , RMSE = 0.2287086

tsdisplay(residuals(model2_Var5), lag.max = 20, main = 'Seasonal Model Residuals')

NOTE: Above Graphs does not shows serious lags

#Plotting forecast for each model

par(mfrow = c(2,2))
fit_auto_train_Var5 %>% forecast(h=341) %>% autoplot() 

model2_Var5 %>% forecast(h=341) %>% autoplot() 

model_Var5 %>% forecast(h=341) %>% autoplot() 

Plot and comment on the residuals of the fitted data for both models in the same plot.

Residual Plot - to confirm no problem with this model

par(mfrow= c(2,2))
hist(fit_auto_train_Var5$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(fit_auto_train_Var5$residuals))

hist(model_Var5$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model_Var5$residuals))

hist(model2_Var5$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model2_Var5$residuals))

OBSERVATION:
The residuals for each model are normally distributed, which shows the good fit of the models. It is normally distrbuted

Testing the autocorrelation and partial autocorrelation of the residuals (using the plots and Ljung-Box Q statistic) up to an appropriate lag *

  1. ACF & PACF Plots
par(mfrow = c(3,2))
acf(fit_auto_train_Var5$residuals, main = 'Correlogram')
pacf(fit_auto_train_Var5$residuals, main = 'Partial Corelogram')

acf(model_Var5$residuals, main = 'Correlogram')
pacf(model_Var5$residuals, main = 'Partial Corelogram')

acf(model2_Var5$residuals, main = 'Correlogram')
pacf(model2_Var5$residuals, main = 'Partial Corelogram')

NOTE:
1. Residuals are the difference between actaual and fitted values
2. These blue lines are signficance bounds
3. This graph shows us the autocorelations for insample forecast errors
4. Do not exceed these significance bounds for lags 1 to end

#Ljunx-Box Test to check the autocorelations of the residuals

HO: No serial correlation upto 20 lags (does not exhibit lack of fit of the model)
Ha: Serial corelation is present (the model exhibits lack of fit)

We reject the null hypothesis, if p value is less than 0.05. This test is generally done for ARIMA models

Box.test(model_Var5$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model_Var5$residuals
## X-squared = 10.265, df = 20, p-value = 0.9631
Box.test(model2_Var5$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model2_Var5$residuals
## X-squared = 8.4094, df = 20, p-value = 0.9888

OBSERVATIONS: For above performed Ljunx-Box Test on both arima models, we can say that p-value is more than significance level 0.05, then there is not a statistical significance.
We conclude that there is a liitle evidence of non-zero auto-corelations in the insample forecast errors at lags 1 to 20.
So we fail to rejects the null hypothesis. Hence, we conclude that there is no serial auto corelation present between lags. Hence, both model seems to be good fit.

Validating The Model by testing the model performance with Holdout set

Here, choosing RMSE as evaluation parameter. The lower RMSE indicates a more accurate forecast.

Below, is the comparison of model. We are fitting the model to test set which is the holdout set. And checking the accuracy using RMSE(Root mean square error) evaluation parameter.

accuracy(forecast(fit_auto_train_Var5), test4) ["Test set", "RMSE"]
## [1] 0.2604829
#o/p - 0.2604829
accuracy(forecast(model2_Var5), test4) ["Test set", "RMSE"]
## [1] 0.248869
#o/p - 0.248869
accuracy(forecast(model_Var5), test4) ["Test set", "RMSE"]
## [1] 0.2489681
#O/P - 0.2489681

NOTE:-
Here, the RMSE value is less of ARIMA models compared to exponential model.
The minimum RMSE is of model2_Var5.

COMMENT ON BEST MODEL AND ITS PARAMETERS

We observed the best model to be ARIMA model i.e.“model2_Var5”.

summary(model2_Var5)
## Series: train4 
## ARIMA(5,0,2) with non-zero mean 
## 
## Coefficients:
##          ar1     ar2      ar3     ar4     ar5     ma1      ma2    mean
##       0.4184  0.8105  -0.3610  0.0955  0.0078  0.0255  -0.9012  0.5693
## s.e.  0.0330  0.0375   0.0305  0.0243  0.0227  0.0253   0.0245  0.0208
## 
## sigma^2 estimated as 0.0525:  log likelihood=123.35
## AIC=-228.69   AICc=-228.61   BIC=-177.46
## 
## Training set error measures:
##                        ME      RMSE       MAE       MPE     MAPE      MASE
## Training set 7.119779e-05 0.2287086 0.1830377 -71.09868 94.38878 0.6568503
##                       ACF1
## Training set -0.0001554728

Best Model is ARIMA (5,0,2) ,which means

p = 5
d = 0
q = 2

For the best model, 5 members of lag are used as a predictor.
As the series was stationary so members of differences needed for stationarity is 0.
2 are the number of lagged forecasts errors used in the prediction equation.

AIC = 6173.8, RMSE = 0.9795425 which is minimum as compared to other models.

6. Forecasting on Geopotential height at 850 hpa, it is about the same as height at low altitude (HT85)

#Checking the attributes of time series

attributes(tsHT85)
## $tsp
## [1] 1998.00 2004.94  365.00
## 
## $class
## [1] "ts"

#Plotting the series

plot(tsHT85, main = "Geopotential height at 850 hpa (tsHT85)", col = "orange")

From the plot, we observes that there seems slight upward trend also seems to have seasonlity.

Checking the statistics and visualising the missing data in the series:

plotNA.distribution(tsHT85)

#data is missing at random
As we see here data is missing randomly, so locf and NOCB would seems the good technique to implement here.

#Impute the missing values with “na_locf” function of “imputeTS package” and visualise the imputed values in the time series

tsHT85.imp <- na_locf(tsHT85)
plotNA.imputations(tsHT85, tsHT85.imp)

#Decomposition Of Time Series

decomp5 <- decompose(tsHT85.imp)
plot(decomp5, col = "red")

OBSERVATIONS:
1)There seems to have trend.
2) It seems to have upward trend from 1998 to around year 2000 and from year 2003 to 2005.
3) Also, observes downward trend from year 2000 to 2003.
4) We observe some seasonality in the series. Seasonality describes cyclical effects due to the time of the year.

Let’s explore this seasonality pattern by plotting the decomp$seasonal components such that y-axis represents Seasonality Index and x-axis has month

#Plotting the decomposed series

plot(decomp5$seasonal[1:365], type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")

OBSERVATIONS:
There is increase in the height in mid of the year and a drop in height for starting and end months of a year and we could think the reason because of seasonal change.

Stationarity check of time series

We could check the stationarity of time series by ADF and KPSS Tests:

  1. Augmented Dickey- Fuller(ADF) t-statistic test for unit root
  2. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) for level or trend stationarity
adf.test(tsHT85.imp)
## Warning in adf.test(tsHT85.imp): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  tsHT85.imp
## Dickey-Fuller = -7.7214, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary
#smaller p-value, it is stationary 
kpss.test(tsHT85.imp, null="Trend")
## Warning in kpss.test(tsHT85.imp, null = "Trend"): p-value smaller than printed
## p-value
## 
##  KPSS Test for Trend Stationarity
## 
## data:  tsHT85.imp
## KPSS Trend = 0.33086, Truncation lag parameter = 8, p-value = 0.01
#smaller p-value, hence not stationary

NOTE: From the both tests results, it leads to CASE4, we could recheck by visualising it i.e. by plotting ACF. Second, we can check each for characteristics of stationarity by looking at the autocorrelation functions (ACF) of each signal. For a stationary signal, because we expect no dependence with time, we would expect the ACF to go to 0 for each time lag (τ). Lets visualize the signals and ACFs.

acf(tsHT85.imp,lag.max = length(tsHT85.imp),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

NOTE: There are significant lags which lies outside the significance bound. Hence, series is not stationary.
Now, we will take the difference to make our data stationary.

It seems difference of 1 is sufficient to make the series stationary.

tsHT85.sta <- diff(tsHT85.imp, diff = 1)
acf(tsHT85.sta, lag.max = length(tsHT85.sta),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Hence, after differencing our series is now transformed to stationary time series.

#Visualising the stationary series

plot(tsHT85.sta, col = "red")

#CREATING MODELS AND FORECASTING: USING EXPONENTIAL SMOOTHING AND ARIMA TECHNIQUES FOR TIME SERIES FORECASTING

1.Train - Test Split of the data

train5 <- window(tsHT85.imp, end = c(2004, 3))
test5 <- window(tsHT85.imp, start = c(2004, 4))

#Model 1 : Auto Exponential Smoothening Model

set.seed(301)
fit_auto_train_Var6 <- forecast(train5)
summary(fit_auto_train_Var6)
## 
## Forecast method: STL +  ETS(A,N,N)
## 
## Model Information:
## ETS(A,N,N) 
## 
## Call:
##  ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend) 
## 
##   Smoothing parameters:
##     alpha = 0.9999 
## 
##   Initial states:
##     l = 1592.2734 
## 
##   sigma:  21.4263
## 
##      AIC     AICc      BIC 
## 30316.22 30316.23 30333.30 
## 
## Error measures:
##                      ME     RMSE      MAE         MPE    MAPE      MASE
## Training set -0.0699278 21.41649 15.85792 -0.01493587 1.04192 0.4709139
##                   ACF1
## Training set 0.1682468
## 
## Forecasts:
##           Point Forecast     Lo 80    Hi 80     Lo 95    Hi 95
## 2004.0082       1455.408 1427.9487 1482.866 1413.4128 1497.402
## 2004.0110       1450.174 1411.3435 1489.005 1390.7878 1509.561
## 2004.0137       1449.436 1401.8788 1496.993 1376.7037 1522.168
## 2004.0164       1420.343 1365.4298 1475.257 1336.3603 1504.327
## 2004.0192       1405.574 1344.1787 1466.969 1311.6781 1499.469
## 2004.0219       1424.431 1357.1765 1491.686 1321.5741 1527.288
## 2004.0247       1436.933 1364.2896 1509.576 1325.8347 1548.031
## 2004.0274       1430.573 1352.9143 1508.232 1311.8044 1549.341
## 2004.0301       1420.404 1338.0347 1502.773 1294.4311 1546.377
## 2004.0329       1421.944 1335.1193 1508.769 1289.1571 1554.731
## 2004.0356       1433.895 1342.8330 1524.958 1294.6275 1573.163
## 2004.0384       1433.117 1338.0054 1528.229 1287.6564 1578.578
## 2004.0411       1417.577 1318.5817 1516.572 1266.1768 1568.977
## 2004.0438       1406.612 1303.8799 1509.344 1249.4968 1563.727
## 2004.0466       1404.724 1298.3866 1511.062 1242.0948 1567.354
## 2004.0493       1415.348 1305.5228 1525.173 1247.3848 1583.311
## 2004.0521       1434.186 1320.9813 1547.392 1261.0541 1607.319
## 2004.0548       1455.310 1338.8234 1571.798 1277.1589 1633.462
## 2004.0575       1444.418 1324.7394 1564.098 1261.3852 1627.452
## 2004.0603       1450.817 1328.0286 1573.605 1263.0285 1638.605
## 2004.0630       1445.513 1319.6932 1571.334 1253.0879 1637.939
## 2004.0658       1419.866 1291.0845 1548.647 1222.9119 1616.819
## 2004.0685       1412.971 1281.2959 1544.647 1211.5911 1614.352
## 2004.0712       1427.331 1292.8234 1561.838 1221.6195 1633.042
## 2004.0740       1429.448 1292.1668 1566.729 1219.4946 1639.401
## 2004.0767       1424.141 1284.1411 1564.141 1210.0297 1638.252
## 2004.0795       1439.794 1297.1269 1582.460 1221.6038 1657.983
## 2004.0822       1428.464 1283.1798 1573.749 1206.2708 1650.658
## 2004.0849       1397.517 1249.6609 1545.373 1171.3905 1623.644
## 2004.0877       1416.996 1266.6124 1567.380 1187.0040 1646.988
## 2004.0904       1424.781 1271.9114 1577.651 1190.9871 1658.575
## 2004.0932       1427.872 1272.5560 1583.187 1190.3368 1665.407
## 2004.0959       1429.397 1271.6729 1587.121 1188.1790 1670.615
## 2004.0986       1428.660 1268.5646 1588.756 1183.8150 1673.506
## 2004.1014       1435.094 1272.6612 1597.527 1186.6743 1683.514
## 2004.1041       1438.982 1274.2449 1603.719 1187.0383 1690.926
## 2004.1068       1443.297 1276.2878 1610.307 1187.8783 1698.716
## 2004.1096       1433.449 1264.1974 1602.700 1174.6012 1692.296
## 2004.1123       1414.543 1243.0788 1586.006 1152.3114 1676.774
## 2004.1151       1403.567 1229.9185 1577.215 1137.9947 1669.138
## 2004.1178       1405.912 1230.1064 1581.717 1137.0407 1674.783
## 2004.1205       1420.234 1242.2981 1598.171 1148.1043 1692.365
## 2004.1233       1432.160 1252.1178 1612.202 1156.8093 1707.511
## 2004.1260       1418.327 1236.2034 1600.451 1139.7930 1696.861
## 2004.1288       1413.408 1229.2262 1597.589 1131.7264 1695.089
## 2004.1315       1401.204 1214.9872 1587.421 1116.4099 1685.998
## 2004.1342       1393.419 1205.1886 1581.649 1105.5456 1681.292
## 2004.1370       1405.684 1215.4624 1595.906 1114.7650 1696.604
## 2004.1397       1434.896 1242.7032 1627.090 1140.9623 1728.830
## 2004.1425       1430.689 1236.5445 1624.833 1133.7706 1727.607
## 2004.1452       1422.273 1226.1972 1618.350 1122.4007 1722.146
## 2004.1479       1411.613 1213.6234 1609.602 1108.8142 1714.411
## 2004.1507       1393.133 1193.2494 1593.017 1087.4372 1698.829
## 2004.1534       1393.228 1191.4676 1594.989 1084.6619 1701.795
## 2004.1562       1394.356 1190.7353 1597.976 1082.9452 1705.766
## 2004.1589       1386.343 1180.8796 1591.806 1072.1140 1700.571
## 2004.1616       1403.492 1196.2025 1610.781 1086.4701 1720.514
## 2004.1644       1417.810 1208.7097 1626.909 1098.0189 1737.600
## 2004.1671       1385.675 1174.7798 1596.569 1063.1388 1708.210
## 2004.1699       1394.340 1181.6653 1607.014 1069.0822 1719.597
## 2004.1726       1413.050 1198.6109 1627.490 1085.0936 1741.007
## 2004.1753       1411.571 1195.3811 1627.761 1080.9370 1742.205
## 2004.1781       1389.549 1171.6223 1607.475 1056.2590 1722.839
## 2004.1808       1377.954 1158.3049 1597.603 1042.0296 1713.879
## 2004.1836       1392.794 1171.4356 1614.153 1054.2555 1731.333
## 2004.1863       1446.625 1223.5706 1669.680 1105.4925 1787.758
## 2004.1890       1456.179 1231.4403 1680.917 1112.4710 1799.886
## 2004.1918       1437.790 1211.3805 1664.199 1091.5267 1784.053
## 2004.1945       1433.818 1205.7502 1661.886 1085.0183 1782.618
## 2004.1973       1435.703 1205.9886 1665.418 1084.3850 1787.021
## 2004.2000       1428.783 1197.4338 1660.133 1074.9647 1782.602
## 2004.2027       1407.770 1174.7970 1640.743 1051.4685 1764.072
## 2004.2055       1381.732 1147.1462 1616.317 1022.9641 1740.499
## 2004.2082       1387.471 1151.2846 1623.658 1026.2549 1748.688
## 2004.2110       1396.179 1158.4017 1633.956 1032.5301 1759.828
## 2004.2137       1411.198 1171.8410 1650.555 1045.1329 1777.263
## 2004.2164       1403.995 1163.0686 1644.922 1035.5297 1772.461
## 2004.2192       1410.955 1168.4693 1653.442 1040.1049 1781.806
## 2004.2219       1438.958 1194.9225 1682.994 1065.7379 1812.178
## 2004.2247       1456.357 1210.7818 1701.932 1080.7821 1831.932
## 2004.2274       1456.336 1209.2304 1703.441 1078.4208 1834.251
## 2004.2301       1440.118 1191.4922 1688.744 1059.8775 1820.359
## 2004.2329       1416.225 1166.0880 1666.363 1033.6733 1798.778
## 2004.2356       1429.560 1177.9205 1681.200 1044.7105 1814.410
## 2004.2384       1436.662 1183.5293 1689.796 1049.5287 1823.796
## 2004.2411       1422.209 1167.5913 1676.827 1032.8048 1811.613
## 2004.2438       1427.304 1171.2104 1683.398 1035.6425 1818.966
## 2004.2466       1432.286 1174.7241 1689.847 1038.3793 1826.192
## 2004.2493       1436.858 1177.8377 1695.879 1040.7204 1832.996
## 2004.2521       1416.287 1155.8149 1676.759 1017.9294 1814.644
## 2004.2548       1411.420 1149.5048 1673.335 1010.8554 1811.984
## 2004.2575       1425.014 1161.6636 1688.364 1022.2545 1827.773
## 2004.2603       1441.971 1177.1934 1706.748 1037.0287 1846.913
## 2004.2630       1445.312 1179.1152 1711.510 1038.1990 1852.426
## 2004.2658       1440.089 1172.4791 1707.698 1030.8153 1849.362
## 2004.2685       1423.346 1154.3321 1692.360 1011.9246 1834.768
## 2004.2712       1421.460 1151.0481 1691.871 1007.9009 1835.019
## 2004.2740       1429.641 1157.8387 1701.443 1013.9555 1845.326
## 2004.2767       1423.568 1150.3826 1696.753 1005.7671 1841.368
## 2004.2795       1440.073 1165.5112 1714.634 1020.1672 1859.978
## 2004.2822       1444.156 1168.2252 1720.087 1022.1563 1866.156
## 2004.2849       1438.234 1160.9402 1715.527 1014.1499 1862.317
## 2004.2877       1430.042 1151.3926 1708.691 1003.8845 1856.199
## 2004.2904       1432.066 1152.0669 1712.064 1003.8445 1860.287
## 2004.2932       1439.716 1158.3741 1721.057 1009.4408 1869.991
## 2004.2959       1425.906 1143.2279 1708.584  993.5870 1858.225
## 2004.2986       1424.140 1140.1317 1708.149  989.7866 1858.494
## 2004.3014       1430.689 1145.3561 1716.021  994.3102 1867.067
## 2004.3041       1438.680 1152.0299 1725.331 1000.2863 1877.074
## 2004.3068       1442.240 1154.2778 1730.203 1001.8397 1882.641
## 2004.3096       1435.141 1145.8730 1724.410  992.7436 1877.539
## 2004.3123       1438.742 1148.1737 1729.311  994.3561 1883.128
## 2004.3151       1435.071 1143.2087 1726.934  988.7059 1881.437
## 2004.3178       1427.698 1134.5470 1720.850  979.3621 1876.035
## 2004.3205       1432.201 1137.7663 1726.635  981.9022 1882.499
## 2004.3233       1436.334 1140.6225 1732.046  984.0822 1888.586
## 2004.3260       1435.758 1138.7746 1732.742  981.5611 1889.955
## 2004.3288       1440.038 1141.7878 1738.288  983.9038 1896.172
## 2004.3315       1447.775 1148.2636 1747.286  989.7120 1905.837
## 2004.3342       1446.136 1145.3696 1746.903  986.1532 1906.120
## 2004.3370       1434.986 1132.9690 1737.004  973.0906 1896.882
## 2004.3397       1423.746 1120.4834 1727.009  959.9457 1887.547
## 2004.3425       1415.515 1111.0113 1720.018  949.8170 1881.212
## 2004.3452       1437.902 1132.1629 1743.640  970.3147 1905.488
## 2004.3479       1432.427 1125.4584 1739.396  962.9589 1901.896
## 2004.3507       1424.881 1116.6868 1733.075  953.5385 1896.224
## 2004.3534       1426.570 1117.1555 1735.985  953.3611 1899.780
## 2004.3562       1433.804 1123.1738 1744.435  958.7358 1908.873
## 2004.3589       1423.742 1111.8999 1735.583  946.8209 1900.662
## 2004.3616       1419.810 1106.7624 1732.859  941.0447 1898.576
## 2004.3644       1431.566 1117.3165 1745.816  950.9627 1912.170
## 2004.3671       1436.547 1121.0996 1751.993  954.1120 1918.981
## 2004.3699       1432.103 1115.4636 1748.743  947.8447 1916.362
## 2004.3726       1430.575 1112.7469 1748.402  944.4990 1916.650
## 2004.3753       1430.771 1111.7595 1749.782  942.8850 1918.657
## 2004.3781       1433.621 1113.4303 1753.812  943.9315 1923.311
## 2004.3808       1441.633 1120.2675 1762.999  950.1467 1933.120
## 2004.3836       1441.048 1118.5111 1763.584  947.7706 1934.325
## 2004.3863       1435.054 1111.3506 1758.757  939.9926 1930.115
## 2004.3890       1429.231 1104.3658 1754.096  932.3925 1926.070
## 2004.3918       1435.478 1109.4540 1761.501  936.8676 1934.087
## 2004.3945       1434.082 1106.9039 1761.259  933.7066 1934.456
## 2004.3973       1435.865 1107.5373 1764.192  933.7312 1937.999
## 2004.4000       1438.063 1108.5894 1767.537  934.1766 1941.949
## 2004.4027       1437.732 1107.1161 1768.347  932.0987 1943.365
## 2004.4055       1429.961 1098.2078 1761.715  922.5880 1937.335
## 2004.4082       1427.415 1094.5266 1760.302  918.3064 1936.523
## 2004.4110       1439.029 1105.0111 1773.048  928.1925 1949.866
## 2004.4137       1439.716 1104.5711 1774.861  927.1562 1952.276
## 2004.4164       1436.329 1100.0618 1772.597  922.0525 1950.606
## 2004.4192       1443.213 1105.8260 1780.599  927.2243 1959.201
## 2004.4219       1447.946 1109.4441 1786.448  930.2520 1965.640
## 2004.4247       1445.100 1105.4863 1784.713  925.7057 1964.494
## 2004.4274       1443.249 1102.5275 1783.971  922.1604 1964.338
## 2004.4301       1455.480 1113.6535 1797.306  932.7017 1978.257
## 2004.4329       1470.565 1127.6380 1813.492  946.1034 1995.026
## 2004.4356       1474.906 1130.8814 1818.930  948.7659 2001.046
## 2004.4384       1454.883 1109.7652 1800.002  927.0706 1982.696
## 2004.4411       1448.917 1102.7084 1795.126  919.4366 1978.397
## 2004.4438       1448.324 1101.0289 1795.620  917.1817 1979.467
## 2004.4466       1458.181 1109.8023 1806.561  925.3814 1990.982
## 2004.4493       1464.060 1114.6010 1813.520  929.6083 1998.513
## 2004.4521       1467.944 1117.4072 1818.480  931.8444 2004.043
## 2004.4548       1467.931 1116.3214 1819.541  930.1903 2005.673
## 2004.4575       1473.485 1120.8051 1826.166  934.1074 2012.864
## 2004.4603       1474.441 1120.6939 1828.189  933.4313 2015.451
## 2004.4630       1471.143 1116.3318 1825.955  928.5060 2013.780
## 2004.4658       1467.304 1111.4318 1823.176  923.0445 2011.563
## 2004.4685       1466.007 1109.0776 1822.937  920.1304 2011.884
## 2004.4712       1466.048 1108.0635 1824.032  918.5582 2013.537
## 2004.4740       1463.287 1104.2514 1822.322  914.1895 2012.384
## 2004.4767       1464.459 1104.3749 1824.542  913.7581 2015.159
## 2004.4795       1473.047 1111.9182 1834.176  920.7480 2025.346
## 2004.4822       1483.708 1121.5366 1845.879  929.8147 2037.601
## 2004.4849       1479.374 1116.1636 1842.585  923.8917 2034.857
## 2004.4877       1471.886 1107.6391 1836.132  914.8185 2028.953
## 2004.4904       1463.629 1098.3485 1828.909  904.9809 2022.276
## 2004.4932       1461.637 1095.3264 1827.947  901.4134 2021.860
## 2004.4959       1463.478 1096.1405 1830.816  901.6836 2025.273
## 2004.4986       1467.187 1098.8243 1835.550  903.8249 2030.549
## 2004.5014       1463.442 1094.0578 1832.827  898.5175 2028.367
## 2004.5041       1466.036 1095.6323 1836.439  899.5526 2032.519
## 2004.5068       1471.486 1100.0662 1842.905  903.4485 2039.523
## 2004.5096       1482.652 1110.2194 1855.086  913.0653 2052.240
## 2004.5123       1478.892 1105.4487 1852.336  907.7595 2050.025
## 2004.5151       1467.885 1093.4338 1842.337  895.2110 2040.560
## 2004.5178       1464.070 1088.6130 1839.527  889.8581 2038.282
## 2004.5205       1466.013 1089.5533 1842.472  890.2677 2041.758
## 2004.5233       1468.248 1090.7883 1845.707  890.9734 2045.522
## 2004.5260       1456.786 1078.3293 1835.242  877.9865 2035.585
## 2004.5288       1449.451 1069.9996 1828.902  869.1302 2029.771
## 2004.5315       1456.095 1075.6521 1836.539  874.2576 2037.933
## 2004.5342       1463.612 1082.1797 1845.045  880.2614 2046.963
## 2004.5370       1474.209 1091.7893 1856.628  889.3486 2059.069
## 2004.5397       1468.815 1085.4115 1852.219  882.4498 2055.181
## 2004.5425       1457.312 1072.9260 1841.698  869.4445 2045.179
## 2004.5452       1455.411 1070.0459 1840.776  866.0459 2044.776
## 2004.5479       1463.888 1077.5457 1850.230  873.0286 2054.747
## 2004.5507       1472.913 1085.5971 1860.230  880.5642 2065.263
## 2004.5534       1461.546 1073.2575 1849.834  867.7101 2055.381
## 2004.5562       1454.605 1065.3477 1843.863  859.2871 2049.924
## 2004.5589       1463.420 1073.1954 1853.645  866.6229 2060.218
## 2004.5616       1471.629 1080.4397 1862.819  873.3565 2069.902
## 2004.5644       1472.404 1080.2524 1864.556  872.6597 2072.149
## 2004.5671       1471.025 1077.9127 1864.136  869.8118 2072.237
## 2004.5699       1474.854 1080.7844 1868.924  872.1766 2077.531
## 2004.5726       1476.485 1081.4597 1871.509  872.3462 2080.623
## 2004.5753       1476.008 1080.0303 1871.986  870.4122 2081.604
## 2004.5781       1475.905 1078.9760 1872.833  868.8547 2082.955
## 2004.5808       1469.166 1071.2894 1867.044  860.6660 2077.667
## 2004.5836       1460.005 1061.1815 1858.828  850.0572 2069.952
## 2004.5863       1467.291 1067.5238 1867.058  855.8998 2078.682
## 2004.5890       1477.914 1077.2050 1878.623  865.0825 2090.745
## 2004.5918       1482.154 1080.5051 1883.802  867.8853 2096.422
## 2004.5945       1475.471 1072.8851 1878.057  859.7690 2091.173
## 2004.5973       1467.428 1063.9070 1870.949  850.2959 2084.560
## 2004.6000       1464.719 1060.2652 1869.173  846.1602 2083.278
## 2004.6027       1464.468 1059.0831 1869.853  844.4854 2084.451
## 2004.6055       1459.530 1053.2162 1865.843  838.1268 2080.933
## 2004.6082       1458.316 1051.0760 1865.556  835.4961 2081.136
## 2004.6110       1463.672 1055.5078 1871.837  839.4385 2087.906
## 2004.6137       1467.345 1058.2579 1876.432  841.7003 2092.990
## 2004.6164       1459.591 1049.5836 1869.598  832.5388 2086.643
## 2004.6192       1449.698 1038.7720 1860.623  821.2411 2078.154
## 2004.6219       1449.224 1037.3819 1861.066  819.3660 2079.082
## 2004.6247       1450.418 1037.6621 1863.174  819.1622 2081.674
## 2004.6274       1443.814 1030.1457 1857.482  811.1630 2076.465
## 2004.6301       1434.443 1019.8643 1849.021  800.3998 2068.486
## 2004.6329       1439.837 1024.3505 1855.324  804.4052 2075.269
## 2004.6356       1442.668 1026.2752 1859.061  805.8502 2079.486
## 2004.6384       1432.493 1015.1964 1849.790  794.2928 2070.694
## 2004.6411       1431.912 1013.7123 1850.111  792.3310 2071.492
## 2004.6438       1450.044 1030.9442 1869.143  809.0863 2091.001
## 2004.6466       1456.178 1036.1802 1876.176  813.8467 2098.510
## 2004.6493       1439.292 1018.3976 1860.186  795.5896 2082.995
## 2004.6521       1433.595 1011.8062 1855.384  788.5246 2078.666
## 2004.6548       1432.217 1009.5355 1854.899  785.7813 2078.653
## 2004.6575       1444.715 1021.1425 1868.287  796.9168 2092.513
## 2004.6603       1447.506 1023.0450 1871.968  798.3487 2096.664
## 2004.6630       1439.861 1014.5122 1865.209  789.3464 2090.375
## 2004.6658       1431.608 1005.3748 1857.842  779.7403 2083.477
## 2004.6685       1441.071 1013.9540 1868.188  787.8519 2094.290
## 2004.6712       1437.083 1009.0845 1865.082  782.5157 2091.651
## 2004.6740       1432.676 1003.7975 1861.554  776.7630 2088.589
## 2004.6767       1433.040 1003.2834 1862.796  775.7841 2090.295
## 2004.6795       1421.810  991.1771 1852.442  763.2140 2080.405
## 2004.6822       1404.419  972.9124 1835.926  744.4864 2064.352
## 2004.6849       1418.693  986.3131 1851.072  757.4252 2079.960
## 2004.6877       1441.328 1008.0779 1874.579  778.7291 2103.927
## 2004.6904       1444.448 1010.3287 1878.568  780.5198 2108.377
## 2004.6932       1441.629 1006.6427 1876.616  776.3746 2106.884
## 2004.6959       1436.454 1000.6017 1872.307  769.8754 2103.033
## 2004.6986       1431.186  994.4699 1867.903  763.2863 2099.086
## 2004.7014       1420.425  982.8461 1858.003  751.2059 2089.643
## 2004.7041       1423.184  984.7446 1861.623  752.6489 2093.718
## 2004.7068       1443.375 1004.0765 1882.672  771.5262 2115.223
## 2004.7096       1444.643 1004.4880 1884.798  771.4840 2117.802
## 2004.7123       1442.310 1001.2991 1883.320  767.8422 2116.777
## 2004.7151       1429.821  987.9563 1871.685  754.0474 2105.594
## 2004.7178       1430.133  987.4165 1872.850  753.0565 2107.210
## 2004.7205       1444.855 1001.2878 1888.422  766.4775 2123.233
## 2004.7233       1450.046 1005.6298 1894.462  770.3701 2129.722
## 2004.7260       1441.737  996.4731 1887.000  760.7649 2122.708
## 2004.7288       1429.017  982.9081 1875.126  746.7521 2111.282
## 2004.7315       1435.649  988.6955 1882.602  752.0927 2119.205
## 2004.7342       1447.085  999.2892 1894.881  762.2404 2131.930
## 2004.7370       1458.632 1009.9952 1907.269  772.5013 2144.763
## 2004.7397       1454.089 1004.6132 1903.565  766.6749 2141.504
## 2004.7425       1456.052 1005.7383 1906.366  767.3565 2144.748
## 2004.7452       1453.218 1002.0679 1904.368  763.2434 2143.193
## 2004.7479       1438.838  986.8531 1890.823  747.5868 2130.089
## 2004.7507       1431.405  978.5875 1884.223  738.8801 2123.931
## 2004.7534       1445.385  991.7356 1899.035  751.5880 2139.182
## 2004.7562       1447.742  993.2627 1902.222  752.6757 2142.809
## 2004.7589       1449.299  993.9903 1904.607  752.9646 2145.632
## 2004.7616       1442.973  986.8375 1899.108  745.3740 2140.572
## 2004.7644       1425.057  968.0956 1882.018  726.1951 2123.918
## 2004.7671       1425.155  967.3695 1882.940  725.0327 2125.276
## 2004.7699       1431.387  972.7793 1889.995  730.0071 2132.767
## 2004.7726       1448.337  988.9081 1907.766  745.7012 2150.973
## 2004.7753       1459.133  998.8848 1919.382  755.2439 2163.023
## 2004.7781       1459.754  998.6871 1920.821  754.6131 2164.894
## 2004.7808       1457.696  995.8129 1919.580  751.3065 2164.086
## 2004.7836       1457.920  995.2211 1920.619  750.2831 2165.557
## 2004.7863       1450.966  987.4532 1914.479  742.0844 2159.847
## 2004.7890       1465.022 1000.6972 1929.348  754.8983 2175.146
## 2004.7918       1479.180 1014.0439 1944.316  767.8157 2190.545
## 2004.7945       1479.152 1013.2061 1945.098  766.5492 2191.755
## 2004.7973       1476.709 1009.9554 1943.464  762.8707 2190.548
## 2004.8000       1465.652  998.0913 1933.213  750.5795 2180.725
## 2004.8027       1458.371  990.0050 1926.738  742.0668 2174.676
## 2004.8055       1470.220 1001.0497 1939.390  752.6858 2187.754
## 2004.8082       1471.542 1001.5694 1941.516  752.7807 2190.304
## 2004.8110       1465.537  994.7629 1936.312  745.5500 2185.525
## 2004.8137       1433.762  962.1878 1905.337  712.5514 2154.973
## 2004.8164       1434.567  962.1940 1906.940  712.1349 2156.999
## 2004.8192       1440.625  967.4543 1913.795  716.9731 2164.276
## 2004.8219       1423.006  949.0400 1896.972  698.1375 2147.875
## 2004.8247       1450.604  975.8435 1925.365  724.5204 2176.688
## 2004.8274       1445.589  970.0350 1921.143  718.2919 2172.886
## 2004.8301       1440.754  964.4083 1917.100  712.2460 2169.262
## 2004.8329       1431.504  954.3675 1908.641  701.7867 2161.221
## 2004.8356       1439.339  961.4132 1917.265  708.4145 2170.263
## 2004.8384       1441.028  962.3142 1919.742  708.8984 2173.158
## 2004.8411       1427.367  947.8664 1906.868  694.0342 2160.700
## 2004.8438       1437.166  956.8801 1917.452  702.6321 2171.700
## 2004.8466       1449.232  968.1622 1930.302  713.4990 2184.966
## 2004.8493       1456.815  974.9622 1938.668  719.8846 2193.746
## 2004.8521       1464.004  981.3698 1946.639  725.8785 2202.130
## 2004.8548       1448.065  964.6505 1931.480  708.7461 2187.385
## 2004.8575       1450.877  966.6831 1935.071  710.3663 2191.388
## 2004.8603       1426.889  941.9170 1911.861  685.1884 2168.589
## 2004.8630       1408.953  923.2046 1894.701  666.0649 2151.841
## 2004.8658       1407.753  921.2294 1894.277  663.6793 2151.827
## 2004.8685       1413.682  926.3840 1900.980  668.4241 2158.939
## 2004.8712       1427.769  939.6982 1915.840  681.3292 2174.209
## 2004.8740       1434.426  945.5838 1923.268  686.8063 2182.046
## 2004.8767       1431.729  942.1166 1921.342  682.9312 2180.527
## 2004.8795       1430.955  940.5730 1921.337  680.9805 2180.930
## 2004.8822       1438.667  947.5173 1929.817  687.5182 2189.817
## 2004.8849       1438.769  946.8522 1930.686  686.4471 2191.091
## 2004.8877       1432.008  939.3256 1924.691  678.5153 2185.501
## 2004.8904       1424.278  930.8308 1917.725  669.6158 2178.940
## 2004.8932       1440.524  946.3137 1934.734  684.6946 2196.353
## 2004.8959       1439.810  944.8382 1934.783  682.8157 2196.805
## 2004.8986       1435.734  940.0012 1931.468  677.5759 2193.893
## 2004.9014       1431.617  935.1245 1928.110  672.2970 2190.938
## 2004.9041       1435.508  938.2563 1932.759  675.0272 2195.988
## 2004.9068       1425.497  927.4879 1923.506  663.8579 2187.136
## 2004.9096       1428.884  930.1191 1927.649  666.0887 2191.680
## 2004.9123       1429.754  930.2338 1929.275  665.8038 2193.705
## 2004.9151       1409.446  909.1715 1909.720  644.3422 2174.549
## 2004.9178       1410.066  909.0385 1911.093  643.8107 2176.321
## 2004.9205       1416.981  915.2024 1918.760  649.5767 2184.386
## 2004.9233       1424.422  921.8923 1926.951  655.8692 2192.974
## 2004.9260       1436.617  933.3382 1939.896  666.9184 2206.316
## 2004.9288       1431.448  927.4209 1935.476  660.6049 2202.292
## 2004.9315       1422.210  917.4350 1926.984  650.2234 2194.196
## 2004.9342       1439.264  933.7435 1944.785  666.1370 2212.392
## 2004.9370       1439.193  932.9276 1945.459  664.9267 2213.460
## 2004.9397       1435.640  928.6304 1942.650  660.2356 2211.045
## 2004.9425       1429.844  922.0915 1937.597  653.3035 2206.385
## 2004.9452       1441.651  933.1566 1950.146  663.9759 2219.326
## 2004.9479       1433.576  924.3411 1942.811  654.7683 2212.384
## 2004.9507       1434.063  924.0883 1944.038  654.1239 2214.002
## 2004.9534       1439.288  928.5748 1950.001  658.2194 2220.357
## 2004.9562       1425.715  914.2641 1937.166  643.5184 2207.912
## 2004.9589       1435.498  923.3111 1947.686  652.1755 2218.821
## 2004.9616       1437.877  924.9548 1950.800  653.4299 2222.325
## 2004.9644       1428.158  914.5009 1941.815  642.5874 2213.728
## 2004.9671       1424.275  909.8846 1938.665  637.5829 2210.967
## 2004.9699       1440.429  925.3070 1955.552  652.6176 2228.241
## 2004.9726       1448.327  932.4730 1964.180  659.3966 2237.257
## 2004.9753       1445.385  928.8009 1961.969  655.3380 2235.432
## 2004.9781       1441.604  924.2909 1958.917  650.4420 2232.766
## 2004.9808       1423.688  905.6473 1941.729  631.4129 2215.964
## 2004.9836       1414.803  896.0349 1933.571  621.4157 2208.190
## 2004.9863       1410.359  890.8650 1929.853  615.8614 2204.857
## 2004.9890       1409.275  889.0556 1929.494  613.6682 2204.881
## 2004.9918       1423.595  902.6521 1944.539  626.8814 2220.309
## 2004.9945       1430.756  909.0897 1952.422  632.9363 2228.576
## 2004.9973       1446.112  923.7239 1968.501  647.1882 2245.036
## 2005.0000       1443.563  920.4535 1966.672  643.5361 2243.590
## 2005.0027       1445.544  921.7143 1969.373  644.4158 2246.672
## 2005.0055       1445.009  920.4606 1969.557  642.7814 2247.237
## 2005.0082       1455.408  930.1410 1980.674  652.0817 2258.733
## 2005.0110       1450.174  924.1907 1976.158  645.7518 2254.597
## 2005.0137       1449.436  922.7361 1976.135  643.9181 2254.953
## 2005.0164       1420.343  892.9286 1947.758  613.7321 2226.955
## 2005.0192       1405.574  877.4446 1933.703  597.8700 2213.277
## 2005.0219       1424.431  895.5889 1953.273  615.6367 2233.226
## 2005.0247       1436.933  907.3783 1966.487  627.0491 2246.816
## 2005.0274       1430.573  900.3072 1960.839  619.6014 2241.544
## 2005.0301       1420.404  889.4279 1951.380  608.3462 2232.462
## 2005.0329       1421.944  890.2585 1953.629  608.8013 2235.087
## 2005.0356       1433.895  901.5016 1966.289  619.6693 2248.122
## 2005.0384       1433.117  900.0155 1966.218  617.8087 2248.425
## 2005.0411       1417.577  883.7689 1951.385  601.1880 2233.966
## 2005.0438       1406.612  872.0984 1941.126  589.1440 2224.080
## 2005.0466       1404.724  869.5061 1939.943  586.1787 2223.270
## 2005.0493       1415.348  879.4258 1951.270  595.7258 2234.970
## 2005.0521       1434.186  897.5615 1970.811  613.4894 2254.883
## 2005.0548       1455.310  917.9836 1992.637  633.5400 2277.081
## 2005.0575       1444.418  906.3906 1982.446  621.5758 2267.261
## 2005.0603       1450.817  912.0887 1989.545  626.9034 2274.730
## 2005.0630       1445.513  906.0863 1984.941  620.5308 2270.496
## 2005.0658       1419.866  879.7402 1959.991  593.8151 2245.916
## 2005.0685       1412.971  872.1485 1953.794  585.8542 2240.089
## 2005.0712       1427.331  885.8116 1968.850  599.1486 2255.513
## 2005.0740       1429.448  887.2330 1971.663  600.2018 2258.694
## 2005.0767       1424.141  881.2313 1967.051  593.8323 2254.450
## 2005.0795       1439.794  896.1902 1983.397  608.4239 2271.163
## 2005.0822       1428.464  884.1681 1972.761  596.0350 2260.894
## 2005.0849       1397.517  852.5287 1942.505  564.0292 2231.005
## 2005.0877       1416.996  871.3166 1962.676  582.4513 2251.541
## 2005.0904       1424.781  878.4112 1971.151  589.1804 2260.382
## 2005.0932       1427.872  880.8124 1974.931  591.2167 2264.527
## 2005.0959       1429.397  881.6489 1977.145  591.6887 2267.105
## 2005.0986       1428.660  880.2247 1977.096  589.9005 2267.420
## 2005.1014       1435.094  885.9718 1984.217  595.2840 2274.904
## 2005.1041       1438.982  889.1737 1988.790  598.1227 2279.841
## 2005.1068       1443.297  892.8038 1993.791  601.3901 2285.204
## 2005.1096       1433.449  882.2710 1984.626  590.4951 2276.402
## 2005.1123       1414.543  862.6815 1966.404  570.5438 2258.541
## 2005.1151       1403.567  851.0228 1956.110  558.5239 2248.609
## 2005.1178       1405.912  852.6863 1959.137  559.8264 2251.997
## 2005.1205       1420.234  866.3281 1974.141  573.1078 2267.361
## 2005.1233       1432.160  877.5736 1986.746  583.9933 2280.327
## 2005.1260       1418.327  863.0614 1973.593  569.1216 2267.533
## 2005.1288       1413.408  857.4638 1969.352  563.1648 2263.651
## 2005.1315       1401.204  844.5824 1957.826  549.9247 2252.483
## 2005.1342       1393.419  836.1202 1950.717  541.1043 2245.733
## 2005.1370       1405.684  847.7100 1963.659  552.3363 2259.032
## 2005.1397       1434.896  876.2470 1993.546  580.5159 2289.277
## 2005.1425       1430.689  871.3652 1990.013  575.2771 2286.101
## 2005.1452       1422.273  862.2763 1982.271  565.8317 2278.715
## 2005.1479       1411.613  850.9428 1972.282  554.1421 2269.083
## 2005.1507       1393.133  831.7916 1954.475  534.6353 2251.631
## 2005.1534       1393.228  831.2157 1955.241  533.7041 2252.753
## 2005.1562       1394.356  831.6726 1957.039  533.8063 2254.905
## 2005.1589       1386.343  822.9903 1949.695  524.7695 2247.916
## 2005.1616       1403.492  839.4708 1967.513  540.8961 2266.088
## 2005.1644       1417.810  853.1206 1982.499  554.1923 2281.427
## 2005.1671       1385.675  820.3184 1951.031  521.0370 2250.312
## 2005.1699       1394.340  828.3174 1960.362  528.6832 2259.996
## 2005.1726       1413.050  846.3625 1979.738  546.3760 2279.725
## 2005.1753       1411.571  844.2184 1978.924  543.8801 2279.262
## 2005.1781       1389.549  821.5321 1957.565  520.8423 2258.255
## 2005.1808       1377.954  809.2744 1946.634  508.2335 2247.675
## 2005.1836       1392.794  823.4520 1962.136  522.0605 2263.528
## 2005.1863       1446.625  876.6216 2016.629  574.8798 2318.371
## 2005.1890       1456.179  885.5139 2026.843  583.4222 2328.935
## 2005.1918       1437.790  866.4649 2009.115  564.0238 2311.556
## 2005.1945       1433.818  861.8339 2005.802  559.0438 2308.592
## 2005.1973       1435.703  863.0605 2008.346  559.9217 2311.485
## 2005.2000       1428.783  855.4828 2002.084  551.9958 2305.571
## 2005.2027       1407.770  833.8125 1981.728  529.9777 2285.563
## 2005.2055       1381.732  807.1176 1956.346  502.9353 2260.528
## 2005.2082       1387.471  812.2017 1962.741  507.6724 2267.270
## 2005.2110       1396.179  820.2545 1972.103  515.3786 2276.979
## 2005.2137       1411.198  834.6196 1987.777  529.3974 2292.999
## 2005.2164       1403.995  826.7634 1981.227  521.1954 2286.795
## 2005.2192       1410.955  833.0710 1988.840  527.1575 2294.753
## 2005.2219       1438.958  860.4218 2017.494  554.1632 2323.753
## 2005.2247       1456.357  877.1696 2035.544  570.5663 2342.148
## 2005.2274       1456.336  876.4979 2036.174  569.5503 2343.121
## 2005.2301       1440.118  859.6306 2020.606  552.3391 2327.897
## 2005.2329       1416.225  835.0889 1997.362  527.4539 2304.997
## 2005.2356       1429.560  847.7756 2011.345  539.7973 2319.323
## 2005.2384       1436.662  854.2302 2019.095  545.9092 2327.416
## 2005.2411       1422.209  839.1301 2005.288  530.4667 2313.952
## 2005.2438       1427.304  843.5792 2011.029  534.5738 2320.035
## 2005.2466       1432.286  847.9152 2016.656  538.5681 2326.003
## 2005.2493       1436.858  851.8434 2021.874  542.1551 2331.562
## 2005.2521       1416.287  830.6277 2001.946  520.5986 2311.975
## 2005.2548       1411.420  825.1174 1997.722  514.7478 2308.092
## 2005.2575       1425.014  838.0688 2011.958  527.3591 2322.668
## 2005.2603       1441.971  854.3843 2029.557  543.3347 2340.607
## 2005.2630       1445.312  857.0847 2033.540  545.6957 2344.929
## 2005.2658       1440.089  851.2204 2028.957  539.4924 2340.685
## 2005.2685       1423.346  833.8384 2012.854  521.7718 2324.921
## 2005.2712       1421.460  831.3129 2011.607  518.9080 2324.012
## 2005.2740       1429.641  838.8554 2020.426  526.1126 2333.169
## 2005.2767       1423.568  832.1448 2014.991  519.0645 2328.071
## 2005.2795       1440.073  848.0128 2032.132  534.5952 2345.550
## 2005.2822       1444.156  851.4598 2036.852  537.7055 2350.607
## 2005.2849       1438.234  844.9019 2031.565  530.8110 2345.656
## 2005.2877       1430.042  836.0753 2024.009  521.6484 2338.436
## 2005.2904       1432.066  837.4647 2026.667  522.7020 2341.429
## 2005.2932       1439.716  844.4813 2034.950  529.3832 2350.048
## 2005.2959       1425.906  830.0388 2021.773  514.6056 2337.207
## 2005.2986       1424.140  827.6406 2020.640  511.8728 2336.408
## 2005.3014       1430.689  833.5576 2027.820  517.4554 2343.922
## 2005.3041       1438.680  840.9184 2036.442  524.4823 2352.878
## 2005.3068       1442.240  843.8479 2040.632  527.0782 2357.402
## 2005.3096       1435.141  836.1195 2034.163  519.0166 2351.266
## 2005.3123       1438.742  839.0915 2038.393  521.6556 2355.829
## 2005.3151       1435.071  834.7925 2035.350  517.0240 2353.119
## 2005.3178       1427.698  826.7918 2028.605  508.6911 2346.705
## 2005.3205       1432.201  830.6672 2033.734  512.2346 2352.167
## 2005.3233       1436.334  834.1744 2038.494  515.4104 2357.258
## 2005.3260       1435.758  832.9729 2038.543  513.8776 2357.639
## 2005.3288       1440.038  836.6275 2043.448  517.2014 2362.874
## 2005.3315       1447.775  843.7401 2051.809  523.9835 2371.566
## 2005.3342       1446.136  841.4782 2050.795  521.3914 2370.881
## 2005.3370       1434.986  829.7051 2040.268  509.2886 2360.684
## 2005.3397       1423.746  817.8426 2029.650  497.0965 2350.396
## 2005.3425       1415.515  808.9891 2022.040  487.9139 2343.115
## 2005.3452       1437.902  830.7549 2045.048  509.3509 2366.452
## 2005.3479       1432.427  824.6602 2040.194  502.9278 2361.927
## 2005.3507       1424.881  816.4942 2033.268  494.4336 2355.329
## 2005.3534       1426.570  817.5642 2035.577  495.1759 2357.965
## 2005.3562       1433.804  824.1797 2043.429  501.4639 2366.145
## 2005.3589       1423.742  813.4989 2033.984  490.4560 2357.027
## 2005.3616       1419.810  808.9504 2030.671  485.5806 2354.040
## 2005.3644       1431.566  820.0896 2043.043  496.3933 2366.739
## 2005.3671       1436.547  824.4536 2048.639  500.4312 2372.662
## 2005.3699       1432.103  819.3948 2044.812  495.0466 2369.160
## 2005.3726       1430.575  817.2514 2043.898  492.5777 2368.572
## 2005.3753       1430.771  816.8335 2044.708  491.8346 2369.707
## 2005.3781       1433.621  819.0700 2048.172  493.7463 2373.496
## 2005.3808       1441.633  826.4692 2056.797  500.8210 2382.446
## 2005.3836       1441.048  825.2711 2056.824  499.2987 2382.797
## 2005.3863       1435.054  818.6653 2051.442  492.3691 2377.738
## 2005.3890       1429.231  812.2316 2046.231  485.6118 2372.850
## 2005.3918       1435.478  817.8674 2053.088  490.9244 2380.031
## 2005.3945       1434.082  815.8614 2052.302  488.5955 2379.568
## 2005.3973       1435.865  817.0354 2054.694  489.4469 2382.283
## 2005.4000       1438.063  818.6247 2057.501  490.7139 2385.412
## 2005.4027       1437.732  817.6852 2057.778  489.4525 2386.011
## 2005.4055       1429.961  809.3075 2050.616  480.7531 2379.170
## 2005.4082       1427.415  806.1535 2048.676  477.2778 2377.551
## 2005.4110       1439.029  817.1619 2060.897  487.9652 2390.094
## 2005.4137       1439.716  817.2426 2062.189  487.7252 2391.707
## 2005.4164       1436.329  813.2509 2059.408  483.4131 2389.246
## 2005.4192       1443.213  819.5294 2066.896  489.3715 2397.054
## 2005.4219       1447.946  823.6588 2072.233  493.1812 2402.711
## 2005.4247       1445.100  820.2092 2069.990  489.4121 2400.788
## 2005.4274       1443.249  817.7556 2068.743  486.6393 2399.859
## 2005.4301       1455.480  829.3837 2081.575  497.9486 2413.010
## 2005.4329       1470.565  843.8674 2097.263  512.1138 2429.016
## 2005.4356       1474.906  847.6070 2102.204  515.5352 2434.276
## 2005.4384       1454.883  826.9841 2082.783  494.5943 2415.172
## 2005.4411       1448.917  820.4177 2077.416  487.7103 2410.124
## 2005.4438       1448.324  819.2258 2077.423  486.2011 2410.448
## 2005.4466       1458.181  828.4839 2087.879  495.1422 2421.221
## 2005.4493       1464.060  833.7646 2094.356  500.1061 2428.015
## 2005.4521       1467.944  837.0500 2098.837  503.0751 2432.812
## 2005.4548       1467.931  836.4407 2099.422  502.1498 2433.713
## 2005.4575       1473.485  841.3982 2105.573  506.7914 2440.180
## 2005.4603       1474.441  841.7581 2107.125  506.8358 2442.047
## 2005.4630       1471.143  837.8644 2104.422  502.6268 2439.660
## 2005.4658       1467.304  833.4303 2101.178  497.8778 2436.730
## 2005.4685       1466.007  831.5392 2100.475  495.6721 2436.342
## 2005.4712       1466.048  830.9858 2101.110  494.8044 2437.291
## 2005.4740       1463.287  827.6318 2098.942  491.1364 2435.437
## 2005.4767       1464.459  828.2109 2100.706  491.4017 2437.516
## 2005.4795       1473.047  836.2073 2109.887  499.0846 2447.010
## 2005.4822       1483.708  846.2763 2121.139  508.8404 2458.575
## 2005.4849       1479.374  841.3516 2117.397  503.6028 2455.145
## 2005.4877       1471.886  833.2728 2110.499  495.2115 2448.560
## 2005.4904       1463.629  824.4256 2102.831  486.0520 2441.205
## 2005.4932       1461.637  821.8445 2101.429  483.1589 2440.115
## 2005.4959       1463.478  823.0973 2103.860  484.0999 2442.857
## 2005.4986       1467.187  826.2174 2108.156  486.9087 2447.465
## 2005.5014       1463.442  821.8849 2104.999  482.2650 2444.619
## 2005.5041       1466.036  823.8912 2108.180  483.9604 2448.111
## 2005.5068       1471.486  828.7545 2114.217  488.5132 2454.458
## 2005.5096       1482.652  839.3351 2125.970  498.7834 2466.522
## 2005.5123       1478.892  834.9894 2122.795  494.1277 2463.657
## 2005.5151       1467.885  823.3973 2112.374  482.2259 2453.545
## 2005.5178       1464.070  818.9972 2109.143  477.5164 2450.623
## 2005.5205       1466.013  820.3560 2111.669  478.5660 2453.460
## 2005.5233       1468.248  822.0074 2114.488  479.9085 2456.587
## 2005.5260       1456.786  809.9626 2103.609  467.5551 2446.017
## 2005.5288       1449.451  802.0451 2096.857  459.3292 2439.572
## 2005.5315       1456.095  808.1077 2104.083  465.0838 2447.107
## 2005.5342       1463.612  815.0433 2112.181  471.7116 2455.513
## 2005.5370       1474.209  825.0589 2123.359  481.4197 2466.998
## 2005.5397       1468.815  819.0852 2118.546  475.1387 2462.492
## 2005.5425       1457.312  807.0016 2107.622  462.7482 2451.875
## 2005.5452       1455.411  804.5215 2106.301  459.9614 2450.861
## 2005.5479       1463.888  812.4193 2115.356  467.5528 2460.222
## 2005.5507       1472.913  820.8668 2124.960  475.6941 2470.133
## 2005.5534       1461.546  808.9213 2114.170  463.4428 2459.649
## 2005.5562       1454.605  801.4037 2107.807  455.6196 2453.591
## 2005.5589       1463.420  809.6418 2117.199  463.5523 2463.288
## 2005.5616       1471.629  817.2745 2125.984  470.8800 2472.379
## 2005.5644       1472.404  817.4737 2127.335  470.7744 2474.034
## 2005.5671       1471.025  815.5188 2126.530  468.5150 2473.534
## 2005.5699       1474.854  818.7734 2130.935  471.4653 2478.243
## 2005.5726       1476.485  819.8298 2133.139  472.2177 2480.751
## 2005.5753       1476.008  818.7796 2133.237  470.8638 2481.152
## 2005.5781       1475.905  818.1028 2133.706  469.8836 2481.926
## 2005.5808       1469.166  810.7919 2127.541  462.2695 2476.063
## 2005.5836       1460.005  801.0580 2118.952  452.2326 2467.777
## 2005.5863       1467.291  807.7725 2126.810  458.6444 2475.938
## 2005.5890       1477.914  817.8242 2138.004  468.3937 2487.434
## 2005.5918       1482.154  821.4930 2142.814  471.7604 2492.547
## 2005.5945       1475.471  814.2400 2136.702  464.2056 2486.736
## 2005.5973       1467.428  805.6274 2129.229  455.2913 2479.565
## 2005.6000       1464.719  802.3493 2127.089  451.7118 2477.727
## 2005.6027       1464.468  801.5292 2127.407  450.5907 2478.345
## 2005.6055       1459.530  796.0227 2123.037  444.7833 2474.276
## 2005.6082       1458.316  794.2413 2122.391  442.7013 2473.931
## 2005.6110       1463.672  799.0302 2128.315  447.1899 2480.155
## 2005.6137       1467.345  802.1358 2132.554  449.9954 2484.694
## 2005.6164       1459.591  793.8155 2125.366  441.3753 2477.807
## 2005.6192       1449.698  783.3563 2116.039  430.6165 2468.779
## 2005.6219       1449.224  782.3170 2116.131  429.2779 2469.170
## 2005.6247       1450.418  782.9464 2117.890  429.6082 2471.228
## 2005.6274       1443.814  775.7778 2111.850  422.1408 2465.487
## 2005.6301       1434.443  765.8426 2103.043  411.9070 2456.978
## 2005.6329       1439.837  770.6734 2109.001  416.4396 2463.235
## 2005.6356       1442.668  772.9413 2112.395  418.4094 2466.927
## 2005.6384       1432.493  762.2042 2102.783  407.3745 2457.612
## 2005.6411       1431.912  761.0602 2102.763  405.9330 2457.890
## 2005.6438       1450.044  778.6308 2121.457  423.2062 2476.881
## 2005.6466       1456.178  784.2041 2128.152  428.4825 2483.874
## 2005.6493       1439.292  766.7573 2111.827  410.7389 2467.845
## 2005.6521       1433.595  760.5002 2106.690  404.1852 2463.005
## 2005.6548       1432.217  758.5624 2105.872  401.9511 2462.483
## 2005.6575       1444.715  770.5009 2118.929  413.5936 2475.836
## 2005.6603       1447.506  772.7336 2122.279  415.5304 2479.482
## 2005.6630       1439.861  764.5295 2115.192  407.0307 2472.691
## 2005.6658       1431.608  755.7193 2107.498  397.9253 2465.292
## 2005.6685       1441.071  764.6245 2117.518  406.5354 2475.607
## 2005.6712       1437.083  760.0796 2114.087  401.6956 2472.471
## 2005.6740       1432.676  755.1159 2110.236  396.4373 2468.914
## 2005.6767       1433.040  754.9236 2111.156  395.9506 2470.129
## 2005.6795       1421.810  743.1379 2100.481  383.8708 2459.748
## 2005.6822       1404.419  725.1924 2083.646  365.6315 2443.207
## 2005.6849       1418.693  738.9110 2098.474  379.0564 2458.329
## 2005.6877       1441.328  760.9925 2121.664  400.8445 2481.812
## 2005.6904       1444.448  763.5586 2125.338  403.1174 2485.779
## 2005.6932       1441.629  760.1866 2123.072  399.4525 2483.806
## 2005.6959       1436.454  754.4583 2118.450  393.4316 2479.477
## 2005.6986       1431.186  748.6381 2113.735  387.3189 2475.054
## 2005.7014       1420.425  737.3244 2103.525  375.7130 2465.136
## 2005.7041       1423.184  739.5319 2106.836  377.6285 2468.739
## 2005.7068       1443.375  759.1716 2127.577  396.9765 2489.773
## 2005.7096       1444.643  759.8896 2129.397  397.4030 2491.883
## 2005.7123       1442.310  757.0059 2127.614  394.2280 2490.391
## 2005.7151       1429.821  743.9672 2115.674  380.8982 2478.743
## 2005.7178       1430.133  743.7303 2116.536  380.3705 2479.896
## 2005.7205       1444.855  757.9032 2131.807  394.2529 2495.457
## 2005.7233       1450.046  762.5457 2137.546  398.6050 2501.487
## 2005.7260       1441.737  753.6883 2129.785  389.4575 2494.016
## 2005.7288       1429.017  740.4213 2117.613  375.9007 2482.134
## 2005.7315       1435.649  746.5058 2124.792  381.6955 2489.602
## 2005.7342       1447.085  757.3952 2136.775  392.2955 2501.875
## 2005.7370       1458.632  768.3959 2148.868  403.0070 2514.257
## 2005.7397       1454.089  763.3074 2144.871  397.6295 2510.549
## 2005.7425       1456.052  764.7248 2147.380  398.7582 2513.346
## 2005.7452       1453.218  761.3457 2145.090  395.0906 2511.345
## 2005.7479       1438.838  746.4210 2131.255  379.8777 2497.798
## 2005.7507       1431.405  738.4444 2124.366  371.6129 2491.198
## 2005.7534       1445.385  751.8804 2138.890  384.7611 2506.009
## 2005.7562       1447.742  753.6944 2141.790  386.2875 2509.197
## 2005.7589       1449.299  754.7076 2143.889  387.0134 2511.584
## 2005.7616       1442.973  747.8395 2138.106  379.8581 2506.088
## 2005.7644       1425.057  729.3812 2120.732  361.1129 2489.000
## 2005.7671       1425.155  728.9375 2121.371  360.3825 2489.926
## 2005.7699       1431.387  734.6288 2128.145  365.7874 2496.987
## 2005.7726       1448.337  751.0380 2145.636  381.9103 2514.764
## 2005.7753       1459.133  761.2940 2156.972  391.8803 2526.386
## 2005.7781       1459.754  761.3747 2158.133  391.6751 2527.833
## 2005.7808       1457.696  758.7777 2156.615  388.7925 2526.600
## 2005.7836       1457.920  758.4622 2157.378  388.1916 2527.648
## 2005.7863       1450.966  750.9695 2150.962  380.4138 2521.518
## 2005.7890       1465.022  764.4877 2165.557  393.6471 2536.398
## 2005.7918       1479.180  778.1076 2180.253  406.9823 2551.378
## 2005.7945       1479.152  777.5420 2180.762  406.1322 2552.172
## 2005.7973       1476.709  774.5625 2178.856  402.8684 2550.550
## 2005.8000       1465.652  762.9687 2168.336  390.9905 2540.314
## 2005.8027       1458.371  755.1516 2161.591  382.8896 2533.853
## 2005.8055       1470.220  766.4646 2173.976  393.9189 2546.521
## 2005.8082       1471.542  767.2516 2175.833  394.4226 2548.662
## 2005.8110       1465.537  760.7115 2170.363  387.5993 2543.475
## 2005.8137       1433.762  728.4017 2139.123  355.0065 2512.518
## 2005.8164       1434.567  728.6724 2140.462  354.9944 2514.140
## 2005.8192       1440.625  734.1962 2147.053  360.2356 2521.013
## 2005.8219       1423.006  716.0444 2129.968  341.8015 2504.211
## 2005.8247       1450.604  743.1096 2158.099  368.5846 2532.624
## 2005.8274       1445.589  737.5618 2153.616  362.7548 2528.423
## 2005.8301       1440.754  732.1948 2149.314  357.1062 2524.402
## 2005.8329       1431.504  722.4129 2140.595  347.0428 2515.965
## 2005.8356       1439.339  729.7166 2148.961  354.0651 2524.613
## 2005.8384       1441.028  730.8746 2151.181  354.9421 2527.114
## 2005.8411       1427.367  716.6830 2138.051  340.4697 2514.264
## 2005.8438       1437.166  725.9520 2148.380  349.4580 2524.874
## 2005.8466       1449.232  737.4884 2160.976  360.7139 2537.751
## 2005.8493       1456.815  744.5419 2169.088  367.4872 2546.143
## 2005.8521       1464.004  751.2022 2176.807  373.8674 2554.141
## 2005.8548       1448.065  734.7345 2161.396  357.1200 2539.011
## 2005.8575       1450.877  737.0181 2164.736  359.1239 2542.630
## 2005.8603       1426.889  712.5020 2141.276  334.3284 2519.449
## 2005.8630       1408.953  694.0387 2123.867  315.5860 2502.320
## 2005.8658       1407.753  692.3119 2123.194  313.5801 2501.926
## 2005.8685       1413.682  697.7139 2129.650  318.7033 2508.660
## 2005.8712       1427.769  711.2747 2144.263  331.9856 2523.552
## 2005.8740       1434.426  717.4061 2151.446  337.8386 2531.014
## 2005.8767       1431.729  714.1838 2149.275  334.3382 2529.121
## 2005.8795       1430.955  712.8844 2149.026  332.7608 2529.149
## 2005.8822       1438.667  720.0720 2157.263  339.6706 2537.664
## 2005.8849       1438.769  719.6494 2157.889  338.9704 2538.568
## 2005.8877       1432.008  712.3645 2151.652  331.4082 2532.608
## 2005.8904       1424.278  704.1106 2144.445  322.8771 2525.678
## 2005.8932       1440.524  719.8335 2161.214  338.3230 2542.725
## 2005.8959       1439.810  718.5973 2161.024  336.8101 2542.811
## 2005.8986       1435.734  713.9987 2157.470  331.9350 2539.534
## 2005.9014       1431.617  709.3598 2153.875  327.0197 2536.215
## 2005.9041       1435.508  712.7284 2158.287  330.1122 2540.903
## 2005.9068       1425.497  702.1962 2148.797  319.3039 2531.690
## 2005.9096       1428.884  705.0627 2152.706  321.8948 2535.874
## 2005.9123       1429.754  705.4121 2154.096  321.9685 2537.540
## 2005.9151       1409.446  684.5835 2134.308  300.8646 2518.027
## 2005.9178       1410.066  684.6836 2135.448  300.6895 2519.442
## 2005.9205       1416.981  691.0798 2142.883  306.8108 2527.152
## 2005.9233       1424.422  698.0012 2150.842  313.4574 2535.386
## 2005.9260       1436.617  709.6779 2163.556  324.8595 2548.375
## 2005.9288       1431.448  703.9907 2158.906  318.8979 2543.999
## 2005.9315       1422.210  694.2341 2150.185  308.8672 2535.552
## 2005.9342       1439.264  710.7712 2167.757  325.1303 2553.398
## 2005.9370       1439.193  710.1832 2168.204  324.2684 2554.119
## 2005.9397       1435.640  706.1130 2165.167  319.9247 2551.356
## 2005.9425       1429.844  699.8006 2159.888  313.3388 2546.350
## 2005.9452       1441.651  711.0914 2172.211  324.3564 2558.946
## 2005.9479       1433.576  702.5008 2164.652  315.4928 2551.660
## 2005.9507       1434.063  702.4722 2165.654  315.1914 2552.935
## 2005.9534       1439.288  707.1822 2171.394  319.6288 2558.947
## 2005.9562       1425.715  693.0944 2158.336  305.2685 2546.161
## 2005.9589       1435.498  702.3635 2168.633  314.2654 2556.731
## 2005.9616       1437.877  704.2286 2171.526  315.8584 2559.896
## 2005.9644       1428.158  693.9955 2162.320  305.3534 2550.962
## 2005.9671       1424.275  689.5992 2158.950  300.6855 2547.864
## 2005.9699       1440.429  705.2410 2175.618  316.0557 2564.803
## 2005.9726       1448.327  712.6257 2184.028  323.1692 2573.484
## 2005.9753       1445.385  709.1716 2181.598  319.4439 2571.326
## 2005.9781       1441.604  704.8789 2178.329  314.8803 2568.327
## 2005.9808       1423.688  686.4520 2160.925  296.1826 2551.194
## 2005.9836       1414.803  677.0556 2152.550  286.5157 2543.090
## 2005.9863       1410.359  672.1010 2148.617  281.2907 2539.428
## 2005.9890       1409.275  670.5063 2148.043  279.4259 2539.124
## 2005.9918       1423.595  684.3168 2162.874  292.9663 2554.224
## 2005.9945       1430.756  690.9678 2170.544  299.3475 2562.164
## 2005.9973       1446.112  705.8147 2186.410  313.9248 2578.300
## 2006.0000       1443.563  702.7563 2184.369  310.5970 2576.529
## 2006.0027       1445.544  704.2286 2186.859  311.8001 2579.287
## 2006.0055       1445.009  703.1856 2186.832  310.4880 2579.530

##NOTE: AIC = 30316.22 , BIC = 30333.30, RMSE = 21.41649 Also, the headline of the plot below contains ETS(A,N,N) showing that the automated model characterized error of the “tsWSR_PK” SERIES as Additative (A), trend as None (A) and seasonality as None (N) which is equivalent to simple exponential model(ses) with only alpha parameter i.e level.

#Model 2 : ARIMA Model It is used for forecasting for both seasonal and non- seasonal level.

#Changing to stationary set

train.sta_Var6 <- diff(train5, diff = 1)
test.sta_Var6 <- diff(test5, diff = 1)

#2.1 Auto Arima model without sesonality:

set.seed(301)
model_Var6 <- auto.arima(train.sta_Var6, seasonal = FALSE)
summary(model_Var6)
## Series: train.sta_Var6 
## ARIMA(3,0,2) with zero mean 
## 
## Coefficients:
##          ar1      ar2      ar3      ma1      ma2
##       0.6344  -0.1583  -0.0120  -0.5604  -0.3554
## s.e.  0.1744   0.1663   0.0809   0.1730   0.1549
## 
## sigma^2 estimated as 446.8:  log likelihood=-9796.41
## AIC=19604.82   AICc=19604.86   BIC=19638.98
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE     MASE        ACF1
## Training set -0.09803186 21.11353 15.43441 NaN  Inf 0.611181 0.001284281

##NOTE: AIC = 19604.82 , BIC = 19638.98, RMSE = 21.11353

tsdisplay(residuals(model_Var6), lag.max = 45, main = '(1,1,1) Model Residuals')

#2.2 Auto Arima model with sesonality:

set.seed(301)
model2_Var6 <- auto.arima(train.sta_Var6, seasonal= TRUE)
summary(model2_Var6)
## Series: train.sta_Var6 
## ARIMA(5,0,3) with non-zero mean 
## 
## Coefficients:
##          ar1      ar2     ar3      ar4     ar5      ma1     ma2      ma3
##       1.0352  -1.4682  1.1148  -0.5114  0.1458  -0.9648  0.9414  -0.8794
## s.e.  0.0732   0.0785  0.0660   0.0343  0.0258   0.0703  0.0722   0.0500
##         mean
##       0.0057
## s.e.  0.0647
## 
## sigma^2 estimated as 447:  log likelihood=-9795.05
## AIC=19610.11   AICc=19610.21   BIC=19667.03
## 
## Training set error measures:
##                      ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set -0.1292709 21.10002 15.42817 NaN  Inf 0.6109338 0.003444275

##NOTE: AIC = 19610.11 , BIC = 19667.03, RMSE = 21.10002

tsdisplay(residuals(model2_Var6), lag.max = 20, main = 'Seasonal Model Residuals')

NOTE: Above Graphs shows no serious lags.

#Plotting forecast for each model

fit_auto_train_Var6 %>% forecast(h=341) %>% autoplot() 

model2_Var6 %>% forecast(h=341) %>% autoplot() 

model_Var6 %>% forecast(h=341) %>% autoplot() 

COMMENT:
1. The plots includes 80% to 95% confidence interval.
2. For ARIMA models it plots the forecast but because of the nature of autoregressive process of 1 lag i.e. AR1, these forecast stabilizes quite smoothly.

Plot and comment on the residuals of the fitted data for both models in the same plot.

Residual Plot - to confirm no problem with this model

par(mfrow=c(1,3))
hist(fit_auto_train_Var6$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(fit_auto_train_Var6$residuals))

hist(model_Var6$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model_Var6$residuals))

hist(model2_Var6$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model2_Var6$residuals))

OBSERVATION: The residuals for each model are normally distributed, which shows the good fit of the models. It is normally distrbuted.

Testing the autocorrelation and partial autocorrelation of the residuals (using the plots and Ljung-Box Q statistic) up to an appropriate lag *

  1. ACF & PACF Plots
par(mfrow = c(3,2))
acf(fit_auto_train_Var6$residuals, main = 'Correlogram')
pacf(fit_auto_train_Var6$residuals, main = 'Partial Corelogram')

acf(model_Var6$residuals, main = 'Correlogram')
pacf(model_Var6$residuals, main = 'Partial Corelogram')

acf(model2_Var6$residuals, main = 'Correlogram')
pacf(model2_Var6$residuals, main = 'Partial Corelogram')

NOTE:
1. Residuals are the difference between actaual and fitted values
2. These blue lines are signficance bounds
3. This graph shows us the autocorelations for insample forecast errors
4. Do not exceed these significance bounds for lags 1 to end

  1. Ljunx-Box Test to check the autocorelations of the residuals

HO: No serial correlation upto 20 lags (does not exhibit lack of fit of the model)
Ha: Serial corelation is present (the model exhibits lack of fit)

We reject the null hypothesis, if p value is less than 0.05. This test is generally done for ARIMA models

Box.test(model_Var6$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model_Var6$residuals
## X-squared = 17.219, df = 20, p-value = 0.6387
Box.test(model2_Var6$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model2_Var6$residuals
## X-squared = 16.943, df = 20, p-value = 0.6567

OBSERVATIONS: For above performed Ljunx-Box Test on both the ARIMA models output, we can say that p-value is more than significance level 0.05, then there is not a statistical significance.
We conclude that there is a little evidence of non-zero auto-corelations in the insample forecast errors at lags 1 to 20.
So we fail to rejects the null hypothesis. Hence, we conclude that there is no serial auto corelation present between lags. Hence, the model seems to be good fit.

Test and comment on the normality of the residuals.

Here, further testing the normality by plotting Normal Q-Q plots. From below plots we observed that most of the datapoints lies on the line. Hence, we could conclude that our model produces residuals that are normal

par(mfrow = c(2,2))
qqnorm(fit_auto_train_Var6$residuals)
qqnorm(model_Var6$residuals)
qqnorm(model2_Var6$residuals)

Validating The Model by testing the model performance with Holdout set

Here, choosing RMSE as evaluation parameter. The lower RMSE indicates a more accurate forecast.

Below, is the comparison of model. We are fitting the model to test set which is the holdout set. And checking the accuracy using RMSE(Root mean square error) evaluation parameter.

accuracy(forecast(fit_auto_train_Var6), test5) ["Test set", "RMSE"]
## [1] 103.0419

#o/p - 103.0419

accuracy(forecast(model2_Var6), test.sta_Var6) ["Test set", "RMSE"]
## [1] 23.82281

#o/p -23.82281

accuracy(forecast(model_Var6), test.sta_Var6) ["Test set", "RMSE"]
## [1] 23.86577

#O/P - 23.86577

NOTE:- Here, the RMSE value is less of ARIMA models compared to exponential model.
The minimum RMSE is of model2_Var6

COMMENT ON BEST MODEL AND ITS PARAMETERS

We observed the best model to be ARIMA model i.e.“model2_Var6”.

summary(model2_Var6)
## Series: train.sta_Var6 
## ARIMA(5,0,3) with non-zero mean 
## 
## Coefficients:
##          ar1      ar2     ar3      ar4     ar5      ma1     ma2      ma3
##       1.0352  -1.4682  1.1148  -0.5114  0.1458  -0.9648  0.9414  -0.8794
## s.e.  0.0732   0.0785  0.0660   0.0343  0.0258   0.0703  0.0722   0.0500
##         mean
##       0.0057
## s.e.  0.0647
## 
## sigma^2 estimated as 447:  log likelihood=-9795.05
## AIC=19610.11   AICc=19610.21   BIC=19667.03
## 
## Training set error measures:
##                      ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set -0.1292709 21.10002 15.42817 NaN  Inf 0.6109338 0.003444275

Passing parameter order = (5,0,3) ,which means

p = 5
d = 0
q = 3 <br.

For the best model, 5 member of lag is used as a predictor. As we transformed the series to stationary so members of differences needed for stationarity is 0. 3 are the number of lagged forecasts errors used in the prediction equation.
AIC = 19610.11, RMSE = 21.10002 which is minimum as compared to other models.

7. Forecasting on T at 700 hpa level (roughly 3100 m height)

#Checking the attributes of time series

attributes(tsT70)
## $tsp
## [1] 1998.00 2004.94  365.00
## 
## $class
## [1] "ts"

#Plotting the series

plot(tsT70, main = "T at 700 hpa level (roughly 3100 m height)(tsT70)", col = "red")

From the plot, we observes that there seems to be no trend but seems to have seasonlity.

Checking the statistics and visualising the missing data in the series:

#Visualizing the series with missing data
plotNA.distribution(tsT70)

As we see the data is missing randomly, so locf and NOCB seems the good technique to implement here.

#Impute the missing values with “na_locf” function of “imputeTS package” and visualise the imputed values in the time series

tsT70.imp <- na_locf(tsT70)
plotNA.imputations(tsT70, tsT70.imp)

#Decomposition Of Time Series

decomp6 <- decompose(tsT70.imp)
plot(decomp6, col = "red")

OBSERVATIONS:
1)There does seems to have trend.
2) We observe the downward trend from year 1999 to 2002.
2)We observe some seasonality in the series. Seasonality describes cyclical effects due to the time of the year.

Let’s explore this seasonality pattern by plotting the decomp$seasonal components such that y-axis represents Seasonality Index and x-axis has month

plot(decomp6$seasonal[1:365], type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")

OBSERVATIONS:
1) There is increase in T for mid month of the year
2) There is a huge drop of T for starting and end months of the year

#Stationarity check of time series

We could check the stationarity of time series by ADF and KPSS Tests:

  1. Augmented Dickey- Fuller(ADF) t-statistic test for unit root
  2. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) for level or trend stationarity
adf.test(tsT70.imp)
## Warning in adf.test(tsT70.imp): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  tsT70.imp
## Dickey-Fuller = -5.8629, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary

##p values is small, hence it is stationary

kpss.test(tsT70.imp, null="Trend")
## 
##  KPSS Test for Trend Stationarity
## 
## data:  tsT70.imp
## KPSS Trend = 0.14889, Truncation lag parameter = 8, p-value = 0.0476

##p - value is smaller than 0.05, hence it is not stationary

NOTE: From the both tests results, it leads to CASE4, we could recheck by visualising it i.e. by plotting ACF. Second, we can check each for characteristics of stationarity by looking at the autocorrelation functions (ACF) of each signal. For a stationary signal, because we expect no dependence with time, we would expect the ACF to go to 0 for each time lag (τ). Lets visualize the signals and ACFs.

Plotting ACF to check the auto-corelations

#Autocorelation plot

acf(tsT70.imp,lag.max = length(tsT70.imp),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Here, we observes most of the data lies outside the blue-dotted lines means the significance level. Now, we will take the difference to make our data stationary.

It seems difference of 1 is sufficient to make the series stationary.

tsT70.sta <- diff(tsT70.imp)
acf(tsT70.sta, lag.max = length(tsT70.sta),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Hence, after differencing our series is now transformed to stationary time series.

#Visualising the stationary series

plot(tsT70.sta, col = "red")

#CREATING MODELS AND FORECASTING: USING EXPONENTIAL SMOOTHING AND ARIMA TECHNIQUES FOR TIME SERIES FORECASTING

  1. TRAIN- TEST SET SPLIT
train6 <- window(tsT70.imp, end = c(2004, 3))
test6 <- window(tsT70.imp, start = c(2004, 4))

#Model 1 : Auto Exponential Smoothening Model

set.seed(301)
fit_auto_train_Var7 <- forecast(train6)
summary(fit_auto_train_Var7)
## 
## Forecast method: STL +  ETS(A,N,N)
## 
## Model Information:
## ETS(A,N,N) 
## 
## Call:
##  ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend) 
## 
##   Smoothing parameters:
##     alpha = 0.8894 
## 
##   Initial states:
##     l = 1.6601 
## 
##   sigma:  1.9024
## 
##      AIC     AICc      BIC 
## 19695.46 19695.47 19712.54 
## 
## Error measures:
##                       ME     RMSE      MAE  MPE MAPE      MASE      ACF1
## Training set 0.002704966 1.901511 1.372217 -Inf  Inf 0.4590085 0.0204421
## 
## Forecasts:
##           Point Forecast       Lo 80     Hi 80      Lo 95     Hi 95
## 2004.0082      3.4821134   1.0441172  5.920110  -0.246480  7.210707
## 2004.0110      2.5256287  -0.7370804  5.788338  -2.464254  7.515512
## 2004.0137      3.7585876  -0.1589008  7.676076  -2.232694  9.749869
## 2004.0164      4.3457573  -0.1317598  8.823274  -2.502014 11.193529
## 2004.0192      3.0227924  -1.9521050  7.997690  -4.585657 10.631242
## 2004.0219      2.3156617  -3.1112205  7.742544  -5.984038 10.615362
## 2004.0247      2.0293226  -3.8146909  7.873336  -6.908325 10.966970
## 2004.0274      4.3840559  -1.8492369 10.617349  -5.148943 13.917055
## 2004.0301      3.9698667  -2.6297836 10.569517  -6.123427 14.063161
## 2004.0329      3.9225818  -3.0241318 10.869295  -6.701500 14.546663
## 2004.0356      2.7692547  -4.5079889 10.046498  -8.360329 13.898838
## 2004.0384      3.6508231  -3.9425765 11.244223  -7.962280 15.263926
## 2004.0411      4.1913478  -3.7055607 12.088256  -7.885932 16.268627
## 2004.0438      4.1592460  -4.0299303 12.348422  -8.365018 16.683510
## 2004.0466      2.5934602  -5.8779064 11.064827 -10.362377 15.549298
## 2004.0493      3.0443987  -5.7000565 11.788854 -10.329092 16.417889
## 2004.0521      3.8863377  -5.1229321 12.895608  -9.892151 17.664827
## 2004.0548      2.6903773  -6.5761425 11.956897 -11.481542 16.862296
## 2004.0575      4.4038388  -5.1129796 13.920657 -10.150879 18.958557
## 2004.0603      3.0452195  -6.7154812 12.805920 -11.882484 17.972923
## 2004.0630      4.1639170  -5.8347190 14.162553 -11.127677 19.455511
## 2004.0658      4.4651485  -5.7658909 14.696188 -11.181876 20.112173
## 2004.0685      2.1753740  -8.2829056 12.633654 -13.819184 18.169932
## 2004.0712      3.2040254  -7.4766607 13.884712 -13.130674 19.538725
## 2004.0740      5.5271570  -5.3713980 16.425712 -11.140745 22.195059
## 2004.0767      4.9326012  -6.1795519 16.044754 -12.061970 21.927173
## 2004.0795      3.6140902  -7.7076319 14.935812 -13.700990 20.929170
## 2004.0822      4.9626588  -6.5648229 16.490141 -12.667103 22.592421
## 2004.0849      5.0456011  -6.6840315 16.775234 -12.893324 22.984526
## 2004.0877      4.4338014  -7.4945567 16.362159 -13.809048 22.676651
## 2004.0904      5.2382856  -6.8855410 17.362112 -13.303507 23.780078
## 2004.0932      4.1948368  -8.1213564 16.511030 -14.641155 23.030829
## 2004.0959      4.4865970  -8.0190042 16.992198 -14.639070 23.612263
## 2004.0986      4.1856260  -8.5065568 16.877809 -15.225393 23.596645
## 2004.1014      5.3155821  -7.5604790 18.191643 -14.376654 25.007818
## 2004.1041      4.2831154  -8.7742348 17.340466 -15.686378 24.252609
## 2004.1068      4.2824310  -8.9537255 17.518588 -15.960524 24.525386
## 2004.1096      2.1809636 -11.2316157 15.593543 -18.331806 22.693734
## 2004.1123      3.3111259 -10.2755856 16.897837 -17.467956 24.090208
## 2004.1151      4.9874704  -8.7711696 18.746110 -16.054554 26.029494
## 2004.1178      5.2870935  -8.6413529 19.215540 -16.014627 26.588814
## 2004.1205      4.3670994  -9.7291080 18.463307 -17.191189 25.925388
## 2004.1233      3.8979611 -10.3640341 18.159956 -17.913878 25.709801
## 2004.1260      3.5132505 -10.9126274 17.939128 -18.549226 25.575727
## 2004.1288      4.0474341 -10.5404855 18.635354 -18.262864 26.357732
## 2004.1315      4.7928847  -9.9552964 19.541066 -17.762512 27.348281
## 2004.1342      4.2468832 -10.6598363 19.153603 -18.550977 27.044743
## 2004.1370      0.9770956 -14.0864940 16.040685 -22.060677 24.014868
## 2004.1397      2.8484039 -12.3704390 18.067247 -20.426808 26.123615
## 2004.1425      3.3846878 -11.9878404 18.757216 -20.125565 26.894941
## 2004.1452      3.6441420 -11.8805501 19.168834 -20.098826 27.387110
## 2004.1479      4.8681392 -10.8072399 20.543518 -19.105284 28.841563
## 2004.1507      4.3069849 -11.5176464 20.131616 -19.894700 28.508670
## 2004.1534      3.7691741 -12.2033148 19.741663 -20.658640 28.196988
## 2004.1562      2.5930154 -13.5259749 18.712006 -22.058853 27.244884
## 2004.1589      2.4611128 -13.8030592 18.725285 -22.412792 27.335017
## 2004.1616      4.5474117 -11.8606576 20.955481 -20.546565 29.641388
## 2004.1644      2.7180617 -13.8326537 19.268777 -22.594073 28.030197
## 2004.1671      4.5195432 -12.1725994 21.211686 -21.008886 30.047972
## 2004.1699      5.7783941 -11.0539875 22.610776 -19.964512 31.521300
## 2004.1726      5.4957186 -11.4757432 22.467180 -20.459892 31.451330
## 2004.1753      5.1143727 -11.9950388 22.223784 -21.052214 31.280959
## 2004.1781      5.4635696 -11.7826881 22.709827 -20.912305 31.839445
## 2004.1808      2.3467987 -15.0352279 19.728825 -24.236717 28.930314
## 2004.1836      2.0917731 -15.4249701 19.608516 -24.697774 28.881320
## 2004.1863      1.1602201 -16.4902116 18.810652 -25.833786 28.154226
## 2004.1890      4.5096232 -13.2734920 22.292738 -22.687304 31.706551
## 2004.1918      5.9831642 -11.9316517 23.897980 -21.415182 33.381511
## 2004.1945      6.3254152 -11.7201403 24.370971 -21.272880 33.923711
## 2004.1973      4.7334749 -13.4418798 22.908830 -23.063331 32.530281
## 2004.2000      4.0451410 -14.2590925 22.349375 -23.948768 32.039050
## 2004.2027      4.9730091 -13.4592020 23.405220 -23.216625 33.162644
## 2004.2055      6.5103089 -12.0489975 25.069615 -21.873701 34.894319
## 2004.2082      6.2915953 -12.3939419 24.977132 -22.285468 34.868658
## 2004.2110      3.5892818 -15.2216390 22.400203 -25.179539 32.358103
## 2004.2137      4.9373643 -13.9981100 23.872839 -24.021945 33.896673
## 2004.2164      5.7682546 -13.2909593 24.827468 -23.380298 34.916807
## 2004.2192      5.9599906 -13.2221646 25.142146 -23.376584 35.296565
## 2004.2219      3.9184128 -15.3859008 23.222726 -25.604987 33.441813
## 2004.2247      5.4870188 -13.9386849 24.912723 -24.222031 35.196069
## 2004.2274      6.5354177 -13.0109225 26.081758 -23.358130 36.428965
## 2004.2301      6.4289092 -13.2373273 26.095146 -23.648004 36.505822
## 2004.2329      8.0451317 -11.7402746 27.830538 -22.214036 38.304299
## 2004.2356      8.5012418 -11.4026208 28.405104 -21.939089 38.941573
## 2004.2384      9.4622148 -10.5594033 29.483833 -21.158208 40.082637
## 2004.2411      8.8449054 -11.2937796 28.983591 -21.954556 39.644366
## 2004.2438      6.7650464 -13.4900291 27.020122 -24.212418 37.742511
## 2004.2466      7.4315769 -12.9392240 27.802378 -23.722875 38.586028
## 2004.2493      7.3183509 -13.1675215 27.804223 -24.012087 38.648789
## 2004.2521      8.8912614 -11.7090400 29.491563 -22.614181 40.396704
## 2004.2548      8.7565501 -11.9575480 29.470648 -22.922929 40.436029
## 2004.2575      8.4205126 -12.4067605 29.247786 -23.432053 40.273078
## 2004.2603      7.1952933 -13.7445432 28.135130 -24.829423 39.220010
## 2004.2630      8.8721698 -12.1796282 29.923968 -23.323777 41.068116
## 2004.2658      9.8584377 -11.3047293 31.021605 -22.507833 42.224709
## 2004.2685     10.2399790 -11.0339742 31.513932 -22.295725 42.775683
## 2004.2712      8.9237045 -12.4604609 30.307870 -23.780554 41.627963
## 2004.2740      8.5458797 -12.9479328 30.039692 -24.326070 41.417829
## 2004.2767      6.9993236 -14.6035795 28.602227 -26.039466 40.038113
## 2004.2795      6.3824239 -15.3290216 28.093869 -26.822367 39.587214
## 2004.2822      7.4904390 -14.3290089 29.309887 -25.879527 40.860405
## 2004.2849      8.5990955 -13.3278230 30.526014 -24.935232 42.133423
## 2004.2877      8.4638796 -13.5699852 30.497744 -25.234009 42.161768
## 2004.2904      9.8294611 -12.3108335 31.969756 -24.031197 43.690120
## 2004.2932     10.2541953 -11.9920199 32.500410 -23.768455 44.276845
## 2004.2959      9.8972717 -12.4543621 32.248905 -24.286602 44.081146
## 2004.2986      8.9201787 -13.5363789 31.376736 -25.424162 43.264520
## 2004.3014      7.8102626 -14.7507308 30.371256 -26.693799 42.314324
## 2004.3041      6.8926315 -15.7723165 29.557579 -27.770415 41.555678
## 2004.3068      7.6653450 -15.1030830 30.433773 -27.155961 42.486651
## 2004.3096      8.1789631 -14.6924767 31.050403 -26.799886 43.157812
## 2004.3123      9.2212045 -13.7527852 32.195194 -25.914481 44.356890
## 2004.3151      9.0340546 -14.0420293 32.110138 -26.257770 44.325879
## 2004.3178      9.0605773 -14.1171511 32.238306 -26.386699 44.507854
## 2004.3205      8.7228710 -14.5560581 32.001800 -26.879179 44.324921
## 2004.3233      9.5554062 -13.8242854 32.935098 -26.200747 45.311559
## 2004.3260      9.1572267 -14.3227951 32.637249 -26.752368 45.066821
## 2004.3288      7.5535666 -16.0263586 31.133492 -28.508817 43.615950
## 2004.3315      6.9332012 -16.7462058 30.612608 -29.281327 43.147729
## 2004.3342      7.1178009 -16.6606718 30.896274 -29.248235 43.483837
## 2004.3370      8.7953890 -15.0817383 32.672516 -27.721526 45.312304
## 2004.3397      9.0849602 -14.8904157 33.060336 -27.582213 45.752133
## 2004.3425      8.7332752 -15.3399485 32.806499 -28.083543 45.550094
## 2004.3452      9.2313540 -14.9393212 33.402029 -27.734504 46.197212
## 2004.3479      9.0852825 -15.1824530 33.353018 -28.029016 46.199581
## 2004.3507      9.3764745 -14.9879346 33.740884 -27.885674 46.638623
## 2004.3534      9.5793047 -14.8813959 34.040005 -27.830109 46.988718
## 2004.3562      8.8195702 -15.7370444 33.376185 -28.736531 46.375671
## 2004.3589     10.1059410 -14.5462143 34.758096 -27.596277 47.808159
## 2004.3616      9.6015698 -15.1457575 34.348897 -28.246201 47.449341
## 2004.3644      9.0110857 -15.8310489 33.853220 -28.981681 47.003852
## 2004.3671      8.5061213 -16.4304603 33.442703 -29.631089 46.643332
## 2004.3699      8.9143140 -16.1163581 33.944986 -29.366795 47.195423
## 2004.3726      9.6320787 -15.4923315 34.756489 -28.792391 48.056548
## 2004.3753      9.8683409 -15.3494590 35.086141 -28.698956 48.435638
## 2004.3781      8.8422073 -16.4686378 34.153052 -29.867390 47.551804
## 2004.3808      9.6363068 -15.7672426 35.039856 -29.215069 48.487683
## 2004.3836     10.0362880 -15.4596287 35.532205 -28.956352 49.028928
## 2004.3863      9.9514910 -15.6364595 35.539442 -29.181902 49.084884
## 2004.3890      9.6098406 -16.0698140 35.289495 -29.663802 48.883483
## 2004.3918      8.3577870 -17.4132452 34.128819 -31.055606 47.771180
## 2004.3945      8.4181658 -17.4439213 34.280253 -31.134483 47.970815
## 2004.3973      8.8443519 -17.1084706 34.797174 -30.847065 48.535769
## 2004.4000      9.2091634 -16.8340783 35.252405 -30.620538 49.038864
## 2004.4027      9.8409131 -16.2924350 35.974261 -30.126594 49.808420
## 2004.4055      9.6744978 -16.5486471 35.897643 -30.430341 49.779337
## 2004.4082      9.6301029 -16.6825323 35.942738 -30.611600 49.871806
## 2004.4110     10.9915218 -15.4103004 37.393344 -29.386581 51.369624
## 2004.4137     10.4781057 -16.0126033 36.968815 -30.035938 50.992149
## 2004.4164     10.2786718 -16.3006266 36.857970 -30.370857 50.928201
## 2004.4192     10.2211904 -16.4464032 36.888784 -30.563375 51.005755
## 2004.4219      9.5005260 -17.2550715 36.256123 -31.418629 50.419681
## 2004.4247      9.6324162 -17.2108966 36.475729 -31.420888 50.685720
## 2004.4274      9.4938697 -17.4368727 36.424612 -31.693147 50.680886
## 2004.4301      8.6067546 -18.4111345 35.624644 -32.713541 49.927050
## 2004.4329      8.8864575 -18.2182980 35.991213 -32.566689 50.339604
## 2004.4356      9.0560002 -18.1353444 36.247345 -32.529573 50.641573
## 2004.4384      9.3078080 -17.9698507 36.585467 -32.409771 51.025387
## 2004.4411      8.3679678 -18.9957328 35.731668 -33.481201 50.217136
## 2004.4438      9.3940601 -18.0554127 36.843533 -32.586286 51.374406
## 2004.4466      9.1995329 -18.3354449 36.734511 -32.911582 51.310647
## 2004.4493      8.9756158 -18.6446022 36.595834 -33.265862 51.217094
## 2004.4521      8.6017445 -19.1034516 36.306941 -33.769696 50.973185
## 2004.4548      8.6920610 -19.0978533 36.481975 -33.808945 51.193067
## 2004.4575      8.8703828 -19.0039922 36.744758 -33.759795 51.500560
## 2004.4603      8.5144784 -19.4441021 36.473059 -34.244481 51.273437
## 2004.4630      9.1506087 -18.8919245 37.193142 -33.736745 52.037962
## 2004.4658      9.0227440 -19.1034913 37.148979 -33.992621 52.038109
## 2004.4685      9.1610218 -19.0486673 37.370711 -33.981975 52.304018
## 2004.4712      9.9443191 -18.3485776 38.237216 -33.325932 53.214571
## 2004.4740     10.1849243 -18.1909360 38.560785 -33.212209 53.582058
## 2004.4767      9.8216771 -18.6369050 38.280259 -33.701968 53.345322
## 2004.4795     10.2126839 -18.3283801 38.753748 -33.437107 53.862475
## 2004.4822     10.2317930 -18.3915153 38.855101 -33.543780 54.007366
## 2004.4849      9.8872820 -18.8180350 38.592599 -34.013712 53.788276
## 2004.4877     10.2421564 -18.5449357 39.029248 -33.783902 54.268215
## 2004.4904     10.2527978 -18.6158377 39.121433 -33.897970 54.403566
## 2004.4932     10.1538837 -18.7960655 39.103833 -34.121243 54.429010
## 2004.4959     10.2407475 -18.7902877 39.271783 -34.158389 54.639884
## 2004.4986     10.1650815 -18.9468139 39.276977 -34.357720 54.687883
## 2004.5014     10.1277208 -19.0648107 39.320252 -34.518403 54.773845
## 2004.5041     10.0705527 -19.2023929 39.343498 -34.698554 54.839660
## 2004.5068     10.5337643 -18.8193749 39.886904 -34.357988 55.425517
## 2004.5096     10.1784045 -19.2547101 39.611519 -34.835660 55.192469
## 2004.5123     10.2580593 -19.2548138 39.770932 -34.877985 55.394104
## 2004.5151     10.1565566 -19.4358600 39.748973 -35.101139 55.414252
## 2004.5178     10.1087691 -19.5629778 39.780516 -35.270252 55.487790
## 2004.5205     10.0362993 -19.7145664 39.787165 -35.463723 55.536322
## 2004.5233      9.6718450 -20.1579296 39.501620 -35.948858 55.292548
## 2004.5260      9.4063739 -20.5021015 39.314849 -36.334692 55.147440
## 2004.5288      9.3140676 -20.6729020 39.301037 -36.547045 55.175180
## 2004.5315     10.0602552 -20.0050037 40.125514 -35.920590 56.041101
## 2004.5342     10.6482731 -19.4950717 40.791618 -35.451995 56.748541
## 2004.5370     10.2382235 -19.9830055 40.459452 -35.981158 56.457605
## 2004.5397      9.9298008 -20.3691121 40.228714 -36.408388 56.267989
## 2004.5425     10.5654203 -19.8109779 40.941819 -35.891272 57.022112
## 2004.5452     10.7652659 -19.6884205 41.218952 -35.809628 57.340160
## 2004.5479      9.7845167 -20.7462621 40.315296 -36.908280 56.477314
## 2004.5507      9.0395213 -21.5681559 39.647198 -37.770882 55.849924
## 2004.5534      8.5537031 -22.1306797 39.238086 -38.374011 55.481417
## 2004.5562      8.9853516 -21.7755456 39.746249 -38.059381 56.030084
## 2004.5589      9.8705768 -20.9666448 40.707798 -37.290884 57.032037
## 2004.5616      9.6709696 -21.2423881 40.584327 -37.606931 56.948870
## 2004.5644     10.1361772 -20.8531294 41.125484 -37.257877 57.530232
## 2004.5671     10.2615480 -20.8035219 41.326618 -37.248377 57.771473
## 2004.5699     10.2924728 -20.8481761 41.433122 -37.333040 57.917986
## 2004.5726     10.2770604 -20.9389845 41.493105 -37.463761 58.017881
## 2004.5753      9.9876609 -21.3035983 41.278920 -37.868190 57.843512
## 2004.5781     10.4778448 -20.8884484 41.844138 -37.492761 58.448451
## 2004.5808     10.1663966 -21.2747515 41.607545 -37.918690 58.251483
## 2004.5836     10.5867371 -20.9290881 42.102562 -37.612558 58.786032
## 2004.5863      9.8003730 -21.7899529 41.390699 -38.512861 58.113607
## 2004.5890     10.1985698 -21.4660814 41.863221 -38.228335 58.625475
## 2004.5918     10.3443720 -21.3944303 42.083174 -38.195937 58.884681
## 2004.5945     10.4943412 -21.3184395 42.307122 -38.159108 59.147791
## 2004.5973      9.6246529 -22.2619346 41.511240 -39.141674 58.390980
## 2004.6000      9.7902271 -22.1699968 41.750451 -39.088717 58.669171
## 2004.6027      9.8565647 -22.1771262 41.890256 -39.134738 58.847867
## 2004.6055      9.9126685 -22.1943214 42.019658 -39.190735 59.016072
## 2004.6082      9.4206484 -22.7594735 41.600770 -39.794601 58.635898
## 2004.6110     10.2728857 -21.9802024 42.525974 -39.053956 59.599727
## 2004.6137     10.5400599 -21.7858296 42.865949 -38.898122 59.978242
## 2004.6164     10.2966706 -22.1018569 42.695198 -39.252601 59.845942
## 2004.6192      9.8083697 -22.6626331 42.279373 -39.851744 59.468483
## 2004.6219      9.5256661 -23.0176507 42.068983 -40.245042 59.296374
## 2004.6247      9.5814856 -23.0339849 42.196956 -40.299572 59.462543
## 2004.6274      9.6249647 -23.0625001 42.312430 -40.366199 59.616128
## 2004.6301      9.2277738 -23.5315272 41.987075 -40.873254 59.328801
## 2004.6329      9.6776200 -23.1533600 42.508600 -40.533031 59.888271
## 2004.6356      9.5081386 -23.3943642 42.410641 -40.811897 59.828174
## 2004.6384      9.6903988 -23.2834717 42.664269 -40.738784 60.119582
## 2004.6411      9.8481057 -23.1969784 42.893190 -40.689989 60.386201
## 2004.6438      9.8367696 -23.2793749 42.952914 -40.810003 60.483542
## 2004.6466      9.8976129 -23.2894399 43.084666 -40.857604 60.652830
## 2004.6493      9.4477551 -23.8100548 42.705565 -41.415676 60.311186
## 2004.6521     10.1127912 -23.2156255 43.441208 -40.858624 61.084206
## 2004.6548     10.5555863 -22.8432881 43.954461 -40.523584 61.634757
## 2004.6575      9.7489903 -23.7201933 43.218174 -41.437709 60.935689
## 2004.6603      9.9416990 -23.5976465 43.481045 -41.352304 61.235702
## 2004.6630     10.1479831 -23.4613778 43.757344 -41.253099 61.549065
## 2004.6658      9.5498076 -24.1294232 43.229038 -41.958131 61.057746
## 2004.6685      9.4137231 -24.3352329 43.162679 -42.200851 61.028297
## 2004.6712      9.6068129 -24.2117246 43.425350 -42.114177 61.327803
## 2004.6740      9.0548033 -24.8331728 42.942779 -42.772384 60.881990
## 2004.6767      9.5596205 -24.3976522 43.516893 -42.373546 61.492787
## 2004.6795      9.2925485 -24.7338796 43.318977 -42.746383 61.331480
## 2004.6822      9.4992808 -24.5961625 43.594724 -42.645200 61.643762
## 2004.6849      9.6608947 -24.5034244 43.825214 -42.588922 61.910712
## 2004.6877      9.2189491 -25.0141072 43.452005 -43.135993 61.573891
## 2004.6904      8.6059664 -25.6956894 42.907622 -43.853889 61.065822
## 2004.6932      9.7487220 -24.6213964 44.118840 -42.815838 62.313282
## 2004.6959     10.2562615 -24.1821833 44.694706 -42.412795 62.925318
## 2004.6986     10.5491299 -23.9575061 45.055766 -42.224216 63.322475
## 2004.7014      9.5014427 -25.0732500 44.076135 -43.375987 62.378872
## 2004.7041      9.7801473 -24.8624682 44.422763 -43.201161 62.761456
## 2004.7068      8.3193953 -26.3910104 43.029801 -44.765589 61.404380
## 2004.7096      7.4988943 -27.2791693 42.276958 -45.689564 60.687353
## 2004.7123      8.1464772 -26.6991129 42.992067 -45.145254 61.438208
## 2004.7151      9.1944363 -25.7185497 44.107422 -44.200368 62.589241
## 2004.7178     10.1383454 -24.8419068 45.118597 -43.359334 63.636024
## 2004.7205     10.4320637 -24.6153254 45.479453 -43.168293 64.032420
## 2004.7233      9.4879223 -25.6264755 44.602320 -44.214915 63.190759
## 2004.7260      8.7143604 -26.4669183 43.895639 -45.090762 62.519483
## 2004.7288      8.4976575 -26.7503754 43.745690 -45.409557 62.404872
## 2004.7315      7.8734705 -27.4411903 43.188131 -46.135642 61.882583
## 2004.7342      7.4479661 -27.9331970 42.829129 -46.662853 61.558786
## 2004.7370      7.7672379 -27.6803030 43.214779 -46.445098 61.979573
## 2004.7397      7.8308069 -27.6829875 43.344601 -46.482855 62.144469
## 2004.7425      6.6069307 -28.9729940 42.186855 -47.807868 61.021730
## 2004.7452      7.2208239 -28.4251084 42.866756 -47.294925 61.736573
## 2004.7479      8.4275325 -27.2842853 44.139350 -46.188980 63.044045
## 2004.7507      7.7589612 -28.0186208 43.536543 -46.958129 62.476051
## 2004.7534      8.1306220 -27.7126036 43.973848 -46.686861 62.948105
## 2004.7562      8.3014454 -27.6073037 44.210195 -46.616247 63.219138
## 2004.7589      6.5187261 -29.4554273 42.492879 -48.498994 61.536446
## 2004.7616      6.7839993 -29.2554396 42.823438 -48.333566 61.901565
## 2004.7644      7.3716476 -28.7329587 43.476254 -47.845583 62.588878
## 2004.7671      6.5046950 -29.6649614 42.674351 -48.812021 61.821411
## 2004.7699      7.7056608 -28.5289289 43.940250 -47.710362 63.121684
## 2004.7726      8.6912089 -27.6081979 44.990616 -46.823943 64.206361
## 2004.7753      8.6051797 -27.7589287 44.969288 -47.008925 64.219284
## 2004.7781      8.8776953 -27.5509997 45.306390 -46.835186 64.590577
## 2004.7808      8.3342666 -28.1589009 44.827434 -47.477217 64.145750
## 2004.7836      8.1834835 -28.3740426 44.741010 -47.726428 64.093395
## 2004.7863      7.1667784 -29.4549932 43.788550 -48.841388 63.174945
## 2004.7890      7.3760709 -29.3098338 44.061976 -48.730179 63.482320
## 2004.7918      7.4626871 -29.2872387 44.212613 -48.741474 63.666849
## 2004.7945      7.6513501 -29.1624856 44.465186 -48.650553 63.953253
## 2004.7973      6.8843199 -29.9933148 43.761955 -49.515155 63.283795
## 2004.8000      6.3360318 -30.6052917 43.277355 -50.160847 62.832911
## 2004.8027      7.6684639 -29.3364388 44.673367 -48.925651 64.262579
## 2004.8055      7.2126868 -29.8556861 44.281060 -49.478497 63.903871
## 2004.8082      6.7247170 -30.4070176 43.856452 -50.063371 63.512805
## 2004.8110      5.9564544 -31.2385341 43.151443 -50.928372 62.841280
## 2004.8137      6.8514132 -30.4067216 44.109548 -50.129987 63.832813
## 2004.8164      5.2032611 -32.1179132 42.524435 -51.874549 62.281072
## 2004.8192      4.5483747 -32.8357328 41.932482 -52.625684 61.722433
## 2004.8219      4.7023620 -32.7445730 42.149297 -52.567783 61.972507
## 2004.8247      6.1656495 -31.3440077 43.675307 -51.200421 63.531720
## 2004.8274      5.6057306 -31.9665441 43.178005 -51.856105 63.067566
## 2004.8301      5.0979558 -32.5368323 42.732744 -52.459486 62.655397
## 2004.8329      5.2180558 -32.4791420 42.915254 -52.434833 62.870945
## 2004.8356      4.4430907 -33.3164137 42.202595 -53.305088 62.191269
## 2004.8384      5.5972037 -32.2245045 43.418912 -52.246107 63.440515
## 2004.8411      5.4255543 -32.4582557 43.309364 -52.512733 63.363842
## 2004.8438      4.9948131 -32.9509970 42.940623 -53.038295 63.027922
## 2004.8466      4.7252441 -33.2824650 42.732953 -53.402531 62.853019
## 2004.8493      4.3867674 -33.6827399 42.456275 -53.835520 62.609055
## 2004.8521      4.5224429 -33.6087627 42.653648 -53.794204 62.839089
## 2004.8548      4.9704096 -33.2223944 43.163214 -53.440444 63.381263
## 2004.8575      4.2412825 -34.0130209 42.495586 -54.263626 62.746191
## 2004.8603      4.9185991 -33.3971048 43.234303 -53.680213 63.517412
## 2004.8630      4.8126553 -33.5643511 43.189662 -53.879911 63.505222
## 2004.8658      5.8985388 -32.5396721 44.336750 -52.887632 64.684709
## 2004.8685      3.4815656 -35.0177526 41.980884 -55.398061 62.361192
## 2004.8712      4.2936370 -34.2666917 42.853966 -54.679297 63.266571
## 2004.8740      5.1536441 -33.4675987 43.774887 -53.912450 64.219738
## 2004.8767      5.0986342 -33.5834268 43.780695 -54.060473 64.257741
## 2004.8795      3.4980516 -35.2447321 42.240835 -55.753923 62.750026
## 2004.8822      4.3913361 -34.4120752 43.194747 -54.953360 63.736033
## 2004.8849      4.8789595 -33.9849849 43.742904 -54.558314 64.316233
## 2004.8877      4.1923813 -34.7320021 43.116765 -55.337326 63.722088
## 2004.8904      2.1794616 -36.8052670 41.164190 -57.442536 61.801459
## 2004.8932      3.8463571 -35.1986235 42.891338 -55.867788 63.560502
## 2004.8959      4.2514796 -34.8536602 43.356619 -55.554671 64.057630
## 2004.8986      3.6108375 -35.5543689 42.776044 -56.287177 63.508852
## 2004.9014      6.3421416 -32.8830396 45.567323 -53.647596 66.331879
## 2004.9041      4.9167190 -34.3683454 44.201783 -55.164602 64.998040
## 2004.9068      4.4882731 -34.8565834 43.833130 -55.684492 64.661038
## 2004.9096      4.8616170 -34.5429408 44.266175 -55.402454 65.125688
## 2004.9123      5.2652872 -34.1988816 44.729456 -55.089951 65.620525
## 2004.9151      3.2647330 -36.2589570 42.788423 -57.181535 63.711001
## 2004.9178      4.1288065 -35.4543150 43.711928 -56.408354 64.665967
## 2004.9205      5.0326986 -34.6097654 44.675163 -55.595218 65.660616
## 2004.9233      2.9132172 -36.7885007 42.614935 -57.805321 63.631755
## 2004.9260      3.5113647 -36.2495187 43.272248 -57.297659 64.320388
## 2004.9288      4.4125137 -35.4074472 44.232475 -56.486861 65.311889
## 2004.9315      2.3753115 -37.5036396 42.254263 -58.614281 63.364904
## 2004.9342      3.8301764 -36.1076775 43.768030 -57.249500 64.909853
## 2004.9370      2.9886507 -37.0080195 42.985321 -58.180978 64.158279
## 2004.9397      1.4142790 -38.6411210 41.469679 -59.845169 62.673727
## 2004.9425      2.8681940 -37.2458499 42.982238 -58.480942 64.217330
## 2004.9452      3.6526821 -36.5199200 43.825284 -57.786011 65.091375
## 2004.9479      3.3844785 -36.8465966 43.615554 -58.143641 64.912598
## 2004.9507      1.6235840 -38.6658793 41.913047 -59.993833 63.241001
## 2004.9534      2.4069517 -37.9408152 42.754719 -59.299633 64.113536
## 2004.9562      0.8795863 -39.5264001 41.285573 -60.916037 62.675210
## 2004.9589      3.3722012 -37.0919210 43.836323 -58.512333 65.256736
## 2004.9616      2.5871211 -37.9350534 43.109296 -59.386197 64.560439
## 2004.9644      1.6026269 -38.9775169 42.182771 -60.459347 63.664601
## 2004.9671      1.2869734 -39.3510571 41.925004 -60.863531 63.437478
## 2004.9699      1.9267866 -38.7690482 42.622621 -60.312122 64.165695
## 2004.9726      0.1962828 -40.5572742 40.949840 -62.130904 62.523470
## 2004.9753      1.5884800 -39.2227177 42.399678 -60.826861 64.003821
## 2004.9781      3.2312324 -37.6375246 44.099989 -59.272138 65.734602
## 2004.9808      2.3686729 -38.5575624 43.294908 -60.222603 64.959949
## 2004.9836      2.0739974 -38.9096357 43.057631 -60.605061 64.753055
## 2004.9863      1.0336628 -40.0072878 42.074614 -61.733055 63.800380
## 2004.9890     -0.2355325 -41.3337208 40.862656 -63.089787 62.618722
## 2004.9918      1.7310214 -39.4243248 42.886368 -61.210649 64.672692
## 2004.9945      1.5290033 -39.6834216 42.741428 -61.499961 64.557968
## 2004.9973      1.2229817 -40.0464429 42.492406 -61.893157 64.339120
## 2005.0000      3.5028204 -37.8235254 44.829166 -59.700371 66.706012
## 2005.0027      2.5034075 -38.8797811 43.886596 -60.786718 65.793533
## 2005.0055      2.4826142 -38.9573393 43.922568 -60.894326 65.859554
## 2005.0082      3.4821134 -38.0145273 44.978754 -59.981522 66.945749
## 2005.0110      2.5256287 -39.0276219 44.078879 -61.024584 66.075841
## 2005.0137      3.7585876 -37.8511959 45.368371 -59.878085 67.395260
## 2005.0164      4.3457573 -37.3204823 46.011997 -59.377257 68.068772
## 2005.0192      3.0227924 -38.6998270 44.745412 -60.786448 66.832032
## 2005.0219      2.3156617 -39.4632614 44.094585 -61.579687 66.211011
## 2005.0247      2.0293226 -39.8058284 43.864474 -61.952020 66.010665
## 2005.0274      4.3840559 -37.5072475 46.275359 -59.683164 68.451276
## 2005.0301      3.9698667 -37.9775140 45.917247 -60.183116 68.122849
## 2005.0329      3.9225818 -38.0808014 45.925965 -60.316049 68.161213
## 2005.0356      2.7692547 -39.2900563 44.828566 -61.554911 67.093420
## 2005.0384      3.6508231 -38.4643414 45.765988 -60.758763 68.060409
## 2005.0411      4.1913478 -37.9795964 46.362292 -60.303546 68.686241
## 2005.0438      4.1592460 -38.0674041 46.385896 -60.420842 68.739334
## 2005.0466      2.5934602 -39.6888223 44.875743 -62.071711 67.258631
## 2005.0493      3.0443987 -39.2934433 45.382241 -61.705743 67.794540
## 2005.0521      3.8863377 -38.5069908 46.279666 -60.948663 68.721339
## 2005.0548      2.6903773 -39.7583654 45.139120 -62.229372 67.610127
## 2005.0575      4.4038388 -38.1002457 46.907923 -60.600549 69.408226
## 2005.0603      3.0452195 -39.5141349 45.604574 -62.043696 68.134135
## 2005.0630      4.1639170 -38.4506355 46.778470 -61.009417 69.337251
## 2005.0658      4.4651485 -38.2045308 47.134828 -60.792495 69.722792
## 2005.0685      2.1753740 -40.5493610 44.900109 -63.166469 67.517217
## 2005.0712      3.2040254 -39.5756943 45.983745 -62.221910 68.629961
## 2005.0740      5.5271570 -37.3074769 48.361791 -59.982762 71.037077
## 2005.0767      4.9326012 -37.9568767 47.822079 -60.661195 70.526397
## 2005.0795      3.6140902 -39.3301616 46.558342 -62.063475 69.291656
## 2005.0822      4.9626588 -38.0362970 47.961615 -60.798569 70.723887
## 2005.0849      5.0456011 -38.0079893 48.099191 -60.799183 70.890386
## 2005.0877      4.4338014 -38.6743543 47.541957 -61.494434 70.362036
## 2005.0904      5.2382856 -37.9243665 48.400938 -60.773294 71.249866
## 2005.0932      4.1948368 -39.0222429 47.411917 -61.899983 70.289657
## 2005.0959      4.4865970 -38.7848419 47.758036 -61.691358 70.664552
## 2005.0986      4.1856260 -39.1401039 47.511356 -62.075360 70.446612
## 2005.1014      5.3155821 -38.0643708 48.695535 -61.028331 71.659495
## 2005.1041      4.2831154 -39.1509928 47.717224 -62.143621 70.709852
## 2005.1068      4.2824310 -39.2057652 47.770627 -62.227026 70.791888
## 2005.1096      2.1809636 -41.3612533 45.723180 -64.411111 68.773038
## 2005.1123      3.3111259 -40.2850448 46.907297 -63.363463 69.985715
## 2005.1151      4.9874704 -38.6625873 48.637528 -61.769532 71.744473
## 2005.1178      5.2870935 -38.4167849 48.990972 -61.552221 72.126408
## 2005.1205      4.3670994 -39.3905334 48.124732 -62.554425 71.288624
## 2005.1233      3.8979611 -39.9133602 47.709282 -63.105673 70.901595
## 2005.1260      3.5132505 -40.3516936 47.378195 -63.572392 70.598893
## 2005.1288      4.0474341 -39.8710672 47.965936 -63.120118 71.214986
## 2005.1315      4.7928847 -39.1791088 48.764878 -62.456476 72.042245
## 2005.1342      4.2468832 -39.7785373 48.272304 -63.084187 71.577954
## 2005.1370      0.9770956 -43.1016872 45.055879 -66.435585 68.389777
## 2005.1397      2.8484039 -41.2836768 46.980485 -64.645789 70.342597
## 2005.1425      3.3846878 -40.8006264 47.570002 -64.190919 70.960294
## 2005.1452      3.6441420 -40.5943416 47.882626 -64.012780 71.301064
## 2005.1479      4.8681392 -39.4234500 49.159728 -62.870001 72.606280
## 2005.1507      4.3069849 -40.0376464 48.651616 -63.512276 72.126246
## 2005.1534      3.7691741 -40.6284358 48.166784 -64.131111 71.669459
## 2005.1562      2.5930154 -41.8575101 47.043541 -65.388197 70.574228
## 2005.1589      2.4611128 -42.0422652 46.964491 -65.600930 70.523156
## 2005.1616      4.5474117 -40.0087563 49.103580 -63.595367 72.690190
## 2005.1644      2.7180617 -41.8908337 47.326957 -65.505356 70.941480
## 2005.1671      4.5195432 -40.1420173 49.181104 -63.784419 72.823506
## 2005.1699      5.7783941 -38.9357696 50.492558 -62.606018 74.162806
## 2005.1726      5.4957186 -39.2709865 50.262424 -62.969049 73.960486
## 2005.1753      5.1143727 -39.7048121 49.933557 -63.430655 73.659401
## 2005.1781      5.4635696 -39.4080335 50.335173 -63.161625 74.088765
## 2005.1808      2.3467987 -42.5771617 47.270759 -66.358470 71.052067
## 2005.1836      2.0917731 -42.8844835 47.068030 -66.693476 70.877022
## 2005.1863      1.1602201 -43.8682721 46.188712 -67.704916 70.025356
## 2005.1890      4.5096232 -40.5710440 49.590290 -64.435308 73.454554
## 2005.1918      5.9831642 -39.1496176 51.115946 -63.041469 75.007798
## 2005.1945      6.3254152 -38.8594212 51.510252 -62.778829 75.429659
## 2005.1973      4.7334749 -40.5033561 49.970306 -64.450288 73.917238
## 2005.2000      4.0451410 -41.2436250 49.333907 -65.218050 73.308332
## 2005.2027      4.9730091 -40.3676324 50.313651 -64.369518 74.315536
## 2005.2055      6.5103089 -38.8821489 51.902767 -62.911464 75.932082
## 2005.2082      6.2915953 -39.1526196 51.735810 -63.209334 75.792524
## 2005.2110      3.5892818 -41.9066313 49.085195 -65.990713 73.169277
## 2005.2137      4.9373643 -40.6101883 50.484917 -64.721606 74.596335
## 2005.2164      5.7682546 -39.8308791 51.367388 -63.969602 75.506112
## 2005.2192      5.9599906 -39.6906659 51.610647 -63.856664 75.776645
## 2005.2219      3.9184128 -41.7837085 49.620534 -65.976950 73.813776
## 2005.2247      5.4870188 -40.2665092 51.240547 -64.486964 75.461002
## 2005.2274      6.5354177 -39.2694596 52.340295 -63.517097 76.587932
## 2005.2301      6.4289092 -39.4272597 52.285078 -63.702049 76.559868
## 2005.2329      8.0451317 -37.8622715 53.952535 -62.164183 78.254446
## 2005.2356      8.5012418 -37.4573386 54.459822 -61.786341 78.788825
## 2005.2384      9.4622148 -36.5474859 55.471915 -60.903550 79.827980
## 2005.2411      8.8449054 -37.2158588 54.905670 -61.598955 79.288765
## 2005.2438      6.7650464 -39.3467249 52.876818 -63.756822 77.286915
## 2005.2466      7.4315769 -38.7311450 53.594299 -63.168214 78.031368
## 2005.2493      7.3183509 -38.8952654 53.531967 -63.359276 77.995978
## 2005.2521      8.8912614 -37.3731935 55.155716 -61.864116 79.646639
## 2005.2548      8.7565501 -37.5586874 55.071788 -62.076493 79.589593
## 2005.2575      8.4205126 -37.9454520 54.786477 -62.490111 79.331136
## 2005.2603      7.1952933 -39.2213429 53.611929 -63.792826 78.183412
## 2005.2630      8.8721698 -37.5950828 55.339422 -62.193360 79.937700
## 2005.2658      9.8584377 -36.6593761 56.376252 -61.284419 81.001295
## 2005.2685     10.2399790 -36.3283413 56.808299 -60.980121 81.460079
## 2005.2712      8.9237045 -37.6950675 55.542476 -62.373555 80.220964
## 2005.2740      8.5458797 -38.1232894 55.215049 -62.828455 79.920215
## 2005.2767      6.9993236 -39.7201882 53.718835 -64.452004 78.450651
## 2005.2795      6.3824239 -40.3873765 53.152224 -65.145813 77.910661
## 2005.2822      7.4904390 -39.3295959 54.310474 -64.114625 79.095503
## 2005.2849      8.5990955 -38.2711202 55.469311 -63.082714 80.280905
## 2005.2877      8.4638796 -38.4564631 55.384222 -63.294592 80.222351
## 2005.2904      9.8294611 -37.1409552 56.799877 -62.005592 81.664514
## 2005.2932     10.2541953 -36.7662413 57.274632 -61.657357 82.165747
## 2005.2959      9.8972717 -37.1731319 56.967675 -62.090698 81.885242
## 2005.2986      8.9201787 -38.2001390 56.040496 -63.144128 80.984486
## 2005.3014      7.8102626 -39.3599164 54.980442 -64.330301 79.950826
## 2005.3041      6.8926315 -40.3273562 54.112619 -65.324108 79.109371
## 2005.3068      7.6653450 -39.6043988 54.935089 -64.627490 79.958180
## 2005.3096      8.1789631 -39.1404845 55.498411 -64.189887 80.547813
## 2005.3123      9.2212045 -38.1478948 56.590304 -63.223581 81.665990
## 2005.3151      9.0340546 -38.3846444 56.452754 -63.486587 81.554697
## 2005.3178      9.0605773 -38.4076696 56.528824 -63.535842 81.656996
## 2005.3205      8.7228710 -38.7948721 56.240614 -63.949246 81.394988
## 2005.3233      9.5554062 -38.0117815 57.122594 -63.192330 82.303142
## 2005.3260      9.1572267 -38.4593543 56.773808 -63.666050 81.980503
## 2005.3288      7.5535666 -40.1123566 55.219490 -65.345172 80.452305
## 2005.3315      6.9332012 -40.7820131 54.648416 -66.040922 79.907324
## 2005.3342      7.1178009 -40.6466538 54.882256 -65.931629 80.167230
## 2005.3370      8.7953890 -39.0182552 56.609033 -64.329270 81.920047
## 2005.3397      9.0849602 -38.7778230 56.947743 -64.114850 82.284770
## 2005.3425      8.7332752 -39.1785967 56.645147 -64.541610 82.008160
## 2005.3452      9.2313540 -38.7295562 57.192264 -64.118528 82.581236
## 2005.3479      9.0852825 -38.9246159 57.095181 -64.339521 82.510086
## 2005.3507      9.3764745 -38.6823624 57.435311 -64.123174 82.876123
## 2005.3534      9.5793047 -38.5284207 57.687030 -63.995112 83.153722
## 2005.3562      8.8195702 -39.3369941 56.976134 -64.829539 82.468680
## 2005.3589     10.1059410 -38.0994127 58.311295 -63.617786 83.829668
## 2005.3616      9.6015698 -38.6525240 57.855664 -64.196698 83.399838
## 2005.3644      9.0110857 -39.2916991 57.313871 -64.861649 82.883820
## 2005.3671      8.5061213 -39.8453054 56.857548 -65.441005 82.453247
## 2005.3699      8.9143140 -39.4857058 57.314334 -65.107129 82.935756
## 2005.3726      9.6320787 -38.8164853 58.080643 -64.463606 83.727763
## 2005.3753      9.8683409 -38.6287188 58.365401 -64.301511 84.038193
## 2005.3781      8.8422073 -39.7032997 57.387714 -65.401739 83.086153
## 2005.3808      9.6363068 -38.9575991 58.230213 -64.681659 83.954273
## 2005.3836     10.0362880 -38.6059687 58.678545 -64.355624 84.428200
## 2005.3863      9.9514910 -38.7390684 58.642050 -64.514294 84.417276
## 2005.3890      9.6098406 -39.1289738 58.348655 -64.929744 84.149425
## 2005.3918      8.3577870 -40.4292345 57.144809 -66.255524 82.971098
## 2005.3945      8.4181658 -40.4170153 57.253347 -66.268799 83.105130
## 2005.3973      8.8443519 -40.0389414 57.727645 -65.916194 83.604897
## 2005.4000      9.2091634 -39.7221947 58.140521 -65.624891 84.043218
## 2005.4027      9.8409131 -39.1384626 58.820289 -65.066578 84.748404
## 2005.4055      9.6744978 -39.3528486 58.701844 -65.306358 84.655354
## 2005.4082      9.6301029 -39.4451673 58.705373 -65.424046 84.684252
## 2005.4110     10.9915218 -38.1316253 60.114669 -64.135849 86.118892
## 2005.4137     10.4781057 -38.6928718 59.649083 -64.722415 85.678626
## 2005.4164     10.2786718 -38.9400896 59.497433 -64.994928 85.552272
## 2005.4192     10.2211904 -39.0453085 59.487689 -65.125418 85.567798
## 2005.4219      9.5005260 -39.8136643 58.814716 -65.919020 84.920072
## 2005.4247      9.6324162 -39.7294193 58.994252 -65.859997 85.124829
## 2005.4274      9.4938697 -39.9155652 58.903305 -66.071340 85.059079
## 2005.4301      8.6067546 -40.8502338 58.063743 -67.031182 84.244691
## 2005.4329      8.8864575 -40.6180386 58.390954 -66.824136 84.597051
## 2005.4356      9.0560002 -40.4959582 58.607959 -66.727181 84.839181
## 2005.4384      9.3078080 -40.2915673 58.907183 -66.547891 85.163507
## 2005.4411      8.3679678 -41.2787791 58.014715 -67.560179 84.296115
## 2005.4438      9.3940601 -40.3000132 59.088133 -66.606467 85.394587
## 2005.4466      9.1995329 -40.5418218 58.940888 -66.873304 85.272370
## 2005.4493      8.9756158 -40.8129753 58.764207 -67.169463 85.120695
## 2005.4521      8.6017445 -41.2340384 58.437527 -67.615508 84.818997
## 2005.4548      8.6920610 -41.1908690 58.574991 -67.597297 84.981419
## 2005.4575      8.8703828 -41.0596497 58.800415 -67.491012 85.231778
## 2005.4603      8.5144784 -41.4626123 58.491569 -67.918886 84.947843
## 2005.4630      9.1506087 -40.8734959 59.174713 -67.354657 85.655875
## 2005.4658      9.0227440 -41.0483303 59.093818 -67.554356 85.599844
## 2005.4685      9.1610218 -40.9569782 59.279022 -67.487845 85.809888
## 2005.4712      9.9443191 -40.2205628 60.109201 -66.776247 86.664885
## 2005.4740     10.1849243 -40.0267956 60.396644 -66.607274 86.977123
## 2005.4767      9.8216771 -40.4368372 60.080191 -67.042088 86.685442
## 2005.4795     10.2126839 -40.0925812 60.517949 -66.722580 87.147948
## 2005.4822     10.2317930 -40.1201796 60.583766 -66.774904 87.238490
## 2005.4849      9.8872820 -40.5113548 60.285919 -67.190782 86.965346
## 2005.4877     10.2421564 -40.2031015 60.687414 -66.907208 87.391521
## 2005.4904     10.2527978 -40.2390380 60.744634 -66.967801 87.473397
## 2005.4932     10.1538837 -40.3844871 60.692255 -67.137885 87.445652
## 2005.4959     10.2407475 -40.3441156 60.825611 -67.122125 87.603620
## 2005.4986     10.1650815 -40.4662311 60.796394 -67.268829 87.598992
## 2005.5014     10.1277208 -40.5499988 60.805440 -67.377163 87.632605
## 2005.5041     10.0705527 -40.6535314 60.794637 -67.505240 87.646345
## 2005.5068     10.5337643 -40.2366419 61.304171 -67.112872 88.180400
## 2005.5096     10.1784045 -40.6382817 60.995091 -67.539010 87.895819
## 2005.5123     10.2580593 -40.6048648 61.120983 -67.530070 88.046189
## 2005.5151     10.1565566 -40.7525632 61.065677 -67.702223 88.015337
## 2005.5178     10.1087691 -40.8465047 61.064043 -67.820597 88.038136
## 2005.5205     10.0362993 -40.9650867 61.037685 -67.963590 88.036188
## 2005.5233      9.6718450 -41.3756115 60.719302 -68.398503 87.742193
## 2005.5260      9.4063739 -41.6871117 60.499859 -68.734369 87.547117
## 2005.5288      9.3140676 -41.8254055 60.453541 -68.897007 87.525143
## 2005.5315     10.0602552 -41.1251642 61.245675 -68.221088 88.341599
## 2005.5342     10.6482731 -40.5830513 61.879597 -67.703276 88.999822
## 2005.5370     10.2382235 -41.0389649 61.515412 -68.183469 88.659916
## 2005.5397      9.9298008 -41.3932105 61.252812 -68.561972 88.421573
## 2005.5425     10.5654203 -40.8033731 61.934214 -67.996370 89.127210
## 2005.5452     10.7652659 -40.6492688 62.179801 -67.866479 89.397011
## 2005.5479      9.7845167 -41.6757186 61.244752 -68.917122 88.486155
## 2005.5507      9.0395213 -42.4663742 60.545417 -69.731948 87.810991
## 2005.5534      8.5537031 -42.9978120 60.105218 -70.287536 87.394942
## 2005.5562      8.9853516 -42.6117429 60.582446 -69.925595 87.896298
## 2005.5589      9.8705768 -41.7720568 61.513210 -69.110016 88.851169
## 2005.5616      9.6709696 -42.0171630 61.359102 -69.379208 88.721147
## 2005.5644     10.1361772 -41.5974143 61.869769 -68.983523 89.255878
## 2005.5671     10.2615480 -41.5174626 62.040559 -68.927615 89.450711
## 2005.5699     10.2924728 -41.5319170 62.116863 -68.966092 89.551037
## 2005.5726     10.2770604 -41.5926690 62.146790 -69.050845 89.604966
## 2005.5753      9.9876609 -41.9273684 61.902690 -69.409525 89.384847
## 2005.5781     10.4778448 -41.4824450 62.438135 -68.988561 89.944251
## 2005.5808     10.1663966 -41.8391143 62.171907 -69.369169 89.701962
## 2005.5836     10.5867371 -41.4639555 62.637430 -69.017928 90.191402
## 2005.5863      9.8003730 -42.2954623 61.896208 -69.873332 89.474078
## 2005.5890     10.1985698 -41.9423690 62.339509 -69.544115 89.941254
## 2005.5918     10.3443720 -41.8416313 62.530375 -69.467233 90.155977
## 2005.5945     10.4943412 -41.7366877 62.725370 -69.386124 90.374807
## 2005.5973      9.6246529 -42.6513629 61.900669 -70.324614 89.573920
## 2005.6000      9.7902271 -42.5307370 62.111191 -70.227782 89.808236
## 2005.6027      9.8565647 -42.5093089 62.222438 -70.230128 89.943257
## 2005.6055      9.9126685 -42.4980763 62.323413 -70.242649 90.067986
## 2005.6082      9.4206484 -43.0349291 61.876226 -70.803234 89.644531
## 2005.6110     10.2728857 -42.2274863 62.773258 -70.019504 90.565276
## 2005.6137     10.5400599 -42.0050683 63.085188 -69.820779 90.900899
## 2005.6164     10.2966706 -42.2931758 62.886517 -70.132559 90.725900
## 2005.6192      9.8083697 -42.8261569 62.442896 -70.689192 90.305932
## 2005.6219      9.5256661 -43.1535028 62.204835 -71.040170 90.091502
## 2005.6247      9.5814856 -43.1422878 62.305259 -71.052568 90.215539
## 2005.6274      9.6249647 -43.1433754 62.393305 -71.077247 90.327177
## 2005.6301      9.2277738 -43.5850956 62.040643 -71.542540 89.998087
## 2005.6329      9.6776200 -43.1797410 62.534981 -71.160738 90.515978
## 2005.6356      9.5081386 -43.3936766 62.409954 -71.398206 90.414483
## 2005.6384      9.6903988 -43.2558333 62.636631 -71.283876 90.664673
## 2005.6411      9.8481057 -43.1425062 62.838718 -71.194042 90.890253
## 2005.6438      9.8367696 -43.1981848 62.871724 -71.273194 90.946733
## 2005.6466      9.8976129 -43.1816471 62.976873 -71.280110 91.075336
## 2005.6493      9.4477551 -43.6757734 62.571284 -71.797671 90.693181
## 2005.6521     10.1127912 -43.0549690 63.280551 -71.200281 91.425864
## 2005.6548     10.5555863 -42.6563689 63.767541 -70.825076 91.936249
## 2005.6575      9.7489903 -43.5071232 63.005104 -71.699207 91.197187
## 2005.6603      9.9416990 -43.3585361 63.241934 -71.573976 91.457374
## 2005.6630     10.1479831 -43.1963372 63.492303 -71.435115 91.731081
## 2005.6658      9.5498076 -43.8385614 62.938177 -72.100657 91.200272
## 2005.6685      9.4137231 -44.0186584 62.846105 -72.304053 91.131499
## 2005.6712      9.6068129 -43.8695449 63.083171 -72.178219 91.391844
## 2005.6740      9.0548033 -44.4654946 62.575101 -72.797429 90.907035
## 2005.6767      9.5596205 -44.0045815 63.123822 -72.359757 91.478998
## 2005.6795      9.2925485 -44.3155216 62.900619 -72.693920 91.279017
## 2005.6822      9.4992808 -44.1526215 63.151183 -72.554223 91.552785
## 2005.6849      9.6608947 -44.0348040 63.356593 -72.459590 91.781379
## 2005.6877      9.2189491 -44.5205104 62.958409 -72.968462 91.406360
## 2005.6904      8.6059664 -45.1772183 62.389151 -73.648316 90.860249
## 2005.6932      9.7487220 -44.0781524 63.575596 -72.572378 92.069822
## 2005.6959     10.2562615 -43.6142670 64.126790 -72.131602 92.644125
## 2005.6986     10.5491299 -43.3650175 64.463277 -71.905443 93.003703
## 2005.7014      9.5014427 -44.4562883 63.459174 -73.019786 92.022671
## 2005.7041      9.7801473 -44.2211321 63.781427 -72.807683 92.367977
## 2005.7068      8.3193953 -45.7253975 62.364188 -74.334983 90.973773
## 2005.7096      7.4988943 -46.5893768 61.587165 -75.221978 90.219767
## 2005.7123      8.1464772 -45.9852373 62.278192 -74.640836 90.933790
## 2005.7151      9.1944363 -44.9806867 63.369559 -73.659265 92.048137
## 2005.7178     10.1383454 -44.0801515 64.356842 -72.781690 93.058381
## 2005.7205     10.4320637 -43.8297723 64.693900 -72.554253 93.418381
## 2005.7233      9.4879223 -44.8172183 63.793063 -73.564623 92.540468
## 2005.7260      8.7143604 -45.6340502 63.062771 -74.404361 91.833082
## 2005.7288      8.4976575 -45.8939888 62.889304 -74.687187 91.682502
## 2005.7315      7.8734705 -46.5613771 62.308318 -75.377445 91.124386
## 2005.7342      7.4479661 -47.0300484 61.925981 -75.868967 90.764900
## 2005.7370      7.7672379 -46.7539096 62.288385 -75.615662 91.150137
## 2005.7397      7.8308069 -46.7334392 62.395053 -75.618006 91.279620
## 2005.7425      6.6069307 -48.0003802 61.214242 -76.907744 90.121606
## 2005.7452      7.2208239 -47.4295179 61.871166 -76.359661 90.801309
## 2005.7479      8.4275325 -46.2658062 63.120871 -75.218711 92.073776
## 2005.7507      7.7589612 -46.9773406 62.495263 -75.952989 91.470911
## 2005.7534      8.1306220 -46.6486093 62.909853 -75.646983 91.908227
## 2005.7562      8.3014454 -46.5206818 63.123573 -75.541763 92.144654
## 2005.7589      6.5187261 -48.3462634 61.383716 -77.390034 90.427487
## 2005.7616      6.7839993 -48.1238191 61.691818 -77.190262 90.758261
## 2005.7644      7.3716476 -47.5789663 62.322261 -76.668064 91.411359
## 2005.7671      6.5046950 -48.4886811 61.498071 -77.600416 90.609806
## 2005.7699      7.7056608 -47.3304442 62.741766 -76.464798 91.876120
## 2005.7726      8.6912089 -46.3875919 63.770010 -75.544548 92.926966
## 2005.7753      8.6051797 -46.5162838 63.726643 -75.695824 92.906184
## 2005.7781      8.8776953 -46.2863980 64.041789 -75.488505 93.243896
## 2005.7808      8.3342666 -46.8724235 63.540957 -76.097080 92.765613
## 2005.7836      8.1834835 -47.0657706 63.432738 -76.312959 92.679926
## 2005.7863      7.1667784 -48.1250069 62.458564 -77.394710 91.728267
## 2005.7890      7.3760709 -47.9582130 62.710355 -77.250413 92.002555
## 2005.7918      7.4626871 -47.9140627 62.839437 -77.228743 92.154117
## 2005.7945      7.6513501 -47.7678331 63.070533 -77.104977 92.407677
## 2005.7973      6.8843199 -48.5772641 62.345904 -77.936853 91.705493
## 2005.8000      6.3360318 -49.1679208 61.839984 -78.549938 91.222002
## 2005.8027      7.6684639 -47.8778248 63.214753 -77.282254 92.619182
## 2005.8055      7.2126868 -48.3759059 62.801280 -77.802729 92.228103
## 2005.8082      6.7247170 -48.9061475 62.355582 -78.355348 91.804782
## 2005.8110      5.9564544 -49.7166498 61.629559 -79.188211 91.101120
## 2005.8137      6.8514132 -48.8638987 62.566725 -78.357803 92.060630
## 2005.8164      5.2032611 -50.5542264 60.960749 -80.070457 90.476980
## 2005.8192      4.5483747 -51.2512567 60.348006 -80.789797 89.886547
## 2005.8219      4.7023620 -51.1393814 60.544105 -80.700215 90.104939
## 2005.8247      6.1656495 -49.7181742 62.049473 -79.301283 91.632583
## 2005.8274      5.6057306 -50.3201417 61.531603 -79.925510 91.136971
## 2005.8301      5.0979558 -50.8699336 61.065845 -80.497545 90.693456
## 2005.8329      5.2180558 -50.7918191 61.227931 -80.441656 90.877767
## 2005.8356      4.4430907 -51.6087383 60.494920 -81.280784 90.166966
## 2005.8384      5.5972037 -50.4965480 61.690955 -80.190786 91.385194
## 2005.8411      5.4255543 -50.7100888 61.561197 -80.426503 91.277612
## 2005.8438      4.9948131 -51.1826901 61.172316 -80.921264 90.910890
## 2005.8466      4.7252441 -51.4940881 60.944576 -81.254805 90.705293
## 2005.8493      4.3867674 -51.8743626 60.647898 -81.657206 90.430741
## 2005.8521      4.5224429 -51.7804541 60.825340 -81.585407 90.630293
## 2005.8548      4.9704096 -51.3742232 61.315042 -81.201270 91.142089
## 2005.8575      4.2412825 -52.1450553 60.627620 -81.994179 90.476744
## 2005.8603      4.9185991 -51.5094128 61.346611 -81.380598 91.217796
## 2005.8630      4.8126553 -51.6570001 61.282311 -81.550230 91.175541
## 2005.8658      5.8985388 -50.6127293 62.409807 -80.527988 92.325065
## 2005.8685      3.4815656 -53.0712846 60.034416 -83.008555 89.971686
## 2005.8712      4.2936370 -52.3007648 60.888039 -82.260031 90.847305
## 2005.8740      5.1536441 -51.4822788 61.789567 -81.463525 91.770813
## 2005.8767      5.0986342 -51.5787794 61.776048 -81.581990 91.779258
## 2005.8795      3.4980516 -53.2208223 60.216925 -83.245980 90.242084
## 2005.8822      4.3913361 -52.3689677 61.151640 -82.416058 91.198730
## 2005.8849      4.8789595 -51.9227442 61.680663 -81.991750 91.749669
## 2005.8877      4.1923813 -52.6506920 61.035455 -82.741597 91.126360
## 2005.8904      2.1794616 -54.7049513 59.063874 -84.817741 89.176664
## 2005.8932      3.8463571 -53.0793653 60.772080 -83.214022 90.906737
## 2005.8959      4.2514796 -52.7155224 61.218482 -82.872032 91.374991
## 2005.8986      3.6108375 -53.3974142 60.619089 -83.575760 90.797435
## 2005.9014      6.3421416 -50.7073300 63.391613 -80.907496 93.591779
## 2005.9041      4.9167190 -52.1739427 62.007381 -82.395913 92.229351
## 2005.9068      4.4882731 -52.6435490 61.620095 -82.887309 91.863855
## 2005.9096      4.8616170 -52.3113359 62.034570 -82.576869 92.300103
## 2005.9123      5.2652872 -51.9487669 62.479341 -82.236058 92.766632
## 2005.9151      3.2647330 -53.9903929 60.519859 -84.299426 90.828892
## 2005.9178      4.1288065 -53.1673616 61.424975 -83.498121 91.755734
## 2005.9205      5.0326986 -52.3044823 62.369880 -82.656952 92.722350
## 2005.9233      2.9132172 -54.4649474 60.291382 -84.839113 90.665547
## 2005.9260      3.5113647 -53.9077542 60.930484 -84.303600 91.326329
## 2005.9288      4.4125137 -53.0475303 61.872558 -83.465040 92.290068
## 2005.9315      2.3753115 -55.1256285 59.876251 -85.564787 90.315410
## 2005.9342      3.8301764 -53.7116305 61.371983 -84.172423 91.832776
## 2005.9370      2.9886507 -54.5939942 60.571296 -85.076405 91.053706
## 2005.9397      1.4142790 -56.2091749 59.037733 -86.713189 89.541747
## 2005.9425      2.8681940 -54.7960400 60.532428 -85.321642 91.058030
## 2005.9452      3.6526821 -54.0523032 61.357667 -84.599477 91.904841
## 2005.9479      3.3844785 -54.3612293 61.130186 -84.929960 91.698918
## 2005.9507      1.6235840 -56.1628177 59.409986 -86.753091 90.000259
## 2005.9534      2.4069517 -55.4201152 60.234019 -86.031915 90.845819
## 2005.9562      0.8795863 -56.9881172 58.747290 -87.621429 89.380602
## 2005.9589      3.3722012 -54.5361104 61.280513 -85.190919 91.935321
## 2005.9616      2.5871211 -55.3617701 60.536012 -86.038060 91.212302
## 2005.9644      1.6026269 -56.3868155 59.592069 -87.084572 90.289826
## 2005.9671      1.2869734 -56.7429920 59.316939 -87.462200 90.036147
## 2005.9699      1.9267866 -56.1436735 59.997247 -86.884318 90.737891
## 2005.9726      0.1962828 -57.9146437 58.307209 -88.676710 89.069276
## 2005.9753      1.5884800 -56.5628847 59.739845 -87.346358 90.523318
## 2005.9781      3.2312324 -54.9605425 61.423007 -85.765407 92.227872
## 2005.9808      2.3686729 -55.8634841 60.600830 -86.689726 91.427072
## 2005.9836      2.0739974 -56.1985137 60.346509 -87.046118 91.194113
## 2005.9863      1.0336628 -57.2791745 59.346500 -88.148126 90.215452
## 2005.9890     -0.2355325 -58.5886682 58.117603 -89.478952 89.007887
## 2005.9918      1.7310214 -56.6623848 60.124428 -87.573987 91.036030
## 2005.9945      1.5290033 -56.9046456 59.962652 -87.837551 90.895558
## 2005.9973      1.2229817 -57.2508823 59.696846 -88.205076 90.651040
## 2006.0000      3.5028204 -55.0112311 62.016872 -85.986699 92.992340
## 2006.0027      2.5034075 -56.0508038 61.057619 -87.047531 92.054346
## 2006.0055      2.4826142 -56.1117294 61.076958 -87.129701 92.094930

##NOTE: AIC = 19695.46 , BIC = 19712.54 , RMSE = 1.901511

Also, the headline of the plot below contains ETS(A,N,N) showing that the automated model characterized error of the “tsWSR_PK” SERIES as Additative (A), trend as None (A) and seasonality as None (N) which is equivalent to simple exponential model(ses) with only alpha parameter i.e level.

#Model 2 : ARIMA Model
It is used for forecasting for both seasonal and non- seasonal level.

#Changing to stationary set

train.sta_Var7 <- diff(train6)
test.sta_Var7 <- diff(test6)

#2.1 Auto Arima model without sesonality:

set.seed(301)
model_Var7 <- auto.arima(train.sta_Var7, seasonal = FALSE)
summary(model_Var7)
## Series: train.sta_Var7 
## ARIMA(1,0,2) with zero mean 
## 
## Coefficients:
##          ar1      ma1      ma2
##       0.5610  -0.7735  -0.1367
## s.e.  0.0359   0.0392   0.0303
## 
## sigma^2 estimated as 4.081:  log likelihood=-4650.53
## AIC=9309.05   AICc=9309.07   BIC=9331.82
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set 0.005417012 2.018759 1.438273 NaN  Inf 0.6448721 0.001386656

##AIC = 9309.05 , BIC = 9331.82, RMSE = 2.018759

tsdisplay(residuals(model_Var7), lag.max = 45, main = '(1,1,1) Model Residuals')

#2.2 Auto Arima model with sesonality:

set.seed(301)
model2_Var7 <- auto.arima(train.sta_Var7, seasonal= TRUE)
summary(model2_Var7)
## Series: train.sta_Var7 
## ARIMA(4,0,1) with zero mean 
## 
## Coefficients:
##          ar1      ar2      ar3     ar4      ma1
##       0.7052  -0.0995  -0.0066  0.0020  -0.9163
## s.e.  0.0273   0.0264   0.0264  0.0238   0.0170
## 
## sigma^2 estimated as 4.083:  log likelihood=-4650.13
## AIC=9312.26   AICc=9312.3   BIC=9346.42
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE          ACF1
## Training set 0.005187141 2.018396 1.437818 NaN  Inf 0.6446682 -3.497019e-05

##AIC = 9312.26 , BIC = 9346.42, RMSE = 2.018396

tsdisplay(residuals(model2_Var7), lag.max = 40, main = 'Seasonal Model Residuals')

NOTE: Above Graphs does not shows serious lags.

Fit the observed with the predicted values for both models in the same plot.

par(mfrow = c(2,2))
fit_auto_train_Var7 %>% forecast(h=341) %>% autoplot() 

model2_Var7 %>% forecast(h=341) %>% autoplot() 

model_Var7 %>% forecast(h=341) %>% autoplot() 

Plot and comment on the residuals of the fitted data for both models in the same plot.

Residual Plot - to confirm no problem with this model

par(mfrow= c(3,2))
hist(fit_auto_train_Var7$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(fit_auto_train_Var7$residuals))

hist(model_Var7$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model_Var7$residuals))

hist(model2_Var7$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model2_Var7$residuals))

OBSERVATION: The residuals for each model are normally distributed, which shows the good fit of the models. It is normally distrbuted.

Testing the autocorrelation and partial autocorrelation of the residuals (using the plots and Ljung-Box Q statistic) up to an appropriate lag *

#ACF PLOTS & PACF PLOTS

par(mfrow = c(3,2))
acf(fit_auto_train_Var7$residuals, main = 'Correlogram')
pacf(fit_auto_train_Var7$residuals, main = 'Partial Corelogram')

acf(model_Var7$residuals, main = 'Correlogram')
pacf(model_Var7$residuals, main = 'Partial Corelogram')

acf(model2_Var7$residuals, main = 'Correlogram')
pacf(model2_Var7$residuals, main = 'Partial Corelogram')

NOTE:
1. Residuals are the difference between actaual and fitted values
2. These blue lines are signficance bounds
3. This graph shows us the autocorelations for insample forecast errors
. 4. Do not exceed these significance bounds for lags 1 to end

  1. Ljunx-Box Test to check the autocorelations of the residuals

HO: No serial correlation upto 20 lags (does not exhibit lack of fit of the model)
Ha: Serial corelation is present (the model exhibits lack of fit)

We reject the null hypothesis, if p value is less than 0.05. This test is generally done for ARIMA models

Box.test(model_Var7$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model_Var7$residuals
## X-squared = 33.203, df = 20, p-value = 0.03205
Box.test(model2_Var7$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model2_Var7$residuals
## X-squared = 31.847, df = 20, p-value = 0.04496

OBSERVATIONS:
For above performed Ljunx-Box Test on models, there is little statistical significance. We conclude that there is a liitle evidence of non-zero auto-corelations in the insample forecast errors at lags 1 to 20.

Test and comment on the normality of the residuals.

Here, further testing the normality by plotting Normal Q-Q plots. From below plots we observed that most of the datapoints lies on the line. Hence, we could conclude that our model produces residuals that are normal

par(mfrow = c(2,2))
qqnorm(fit_auto_train_Var7$residuals)
qqnorm(model_Var7$residuals)
qqnorm(model2_Var7$residuals)

Validating The Model by testing the model performance with Holdout set

Here, choosing RMSE as evaluation parameter. The lower RMSE indicates a more accurate forecast.

Below, is the comparison of model. We are fitting the model to test set which is the holdout set. And checking the accuracy using RMSE(Root mean square error) evaluation parameter.

accuracy(forecast(fit_auto_train_Var7), test6) ["Test set", "RMSE"]
## [1] 2.870951

#o/p - 2.870951

accuracy(forecast(model2_Var7), test.sta_Var7) ["Test set", "RMSE"]
## [1] 2.00263

#o/p -2.00263

accuracy(forecast(model_Var7), test.sta_Var7) ["Test set", "RMSE"]
## [1] 2.002481

#O/P - 2.002481

NOTE:- Here, the RMSE value is less of ARIMA models compared to exponential model.
The minimum RMSE is of model_Var7

COMMENT ON BEST MODEL AND ITS PARAMETERS

We observed the best model to be ARIMA model i.e.“model_Var7”.

summary(model_Var7)
## Series: train.sta_Var7 
## ARIMA(1,0,2) with zero mean 
## 
## Coefficients:
##          ar1      ma1      ma2
##       0.5610  -0.7735  -0.1367
## s.e.  0.0359   0.0392   0.0303
## 
## sigma^2 estimated as 4.081:  log likelihood=-4650.53
## AIC=9309.05   AICc=9309.07   BIC=9331.82
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set 0.005417012 2.018759 1.438273 NaN  Inf 0.6448721 0.001386656

Passing parameter order = (1,0,2) ,which means

p = 1, d = 0 q = 2

For the best model, 1 member of lag is used as a predictor. As we transformed the series to stationary so members of differences needed for stationarity is 0. 2 are the number of lagged forecasts errors used in the prediction equation. AIC = 9309.07, RMSE = 2.018759 which is minimum as compared to other models.

8. Forecasting on K-Index (KI)

#Checking the attributes of time series

attributes(tsKI)
## $tsp
## [1] 1998.00 2004.94  365.00
## 
## $class
## [1] "ts"

#Plotting the series

plot(tsKI, main = "Average T observed for years (tsKI)", col = "red")

From the plot, we observes that there is no trend but seems to have seasonlity.

Checking the statistics and visualising the missing data in the series:

plotNA.distribution(tsKI)

Here, data is missing at random.

#Impute the missing values with “na_interpolation” function of “imputeTS package” and visualise the imputed values in the time series

tsKI.imp <- na_interpolation(tsKI, method = "linear")
plotNA.imputations(tsKI, tsKI.imp)

#Decomposition Of Time Series

decomp7 <- decompose(tsKI.imp)
plot(decomp7, col = "red")

OBSERVATIONS:
1)There does seems to have trend.
2)We observes the downward trend from year 1999 to mid of year 2001 and then upward trend from year onwards.
3)We observe some seasonality in the series.Seasonality describes cyclical effects due to the time of the year.

Let’s explore this seasonality pattern by plotting the decomp$seasonal components such that y-axis represents Seasonality Index and x-axis has month

#Plotting the decomposed series

plot(decomp7$seasonal[1:365], type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")

OBSERVATIONS:

  1. There is increase in KI for mid month of the year
  2. There is a huge drop of KI for starting and end months of the year

#Stationarity check of time series

We could check the stationarity of time series by ADF and KPSS Tests:

  1. Augmented Dickey- Fuller(ADF) t-statistic test for unit root
  2. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) for level or trend stationarity
adf.test(tsKI.imp)
## Warning in adf.test(tsKI.imp): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  tsKI.imp
## Dickey-Fuller = -5.9654, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary

#p-value is smaller, hence series is stationary

kpss.test(tsKI.imp, null = "Trend")
## 
##  KPSS Test for Trend Stationarity
## 
## data:  tsKI.imp
## KPSS Trend = 0.12165, Truncation lag parameter = 8, p-value = 0.09509

#series is not stationary

acf(tsKI.imp,lag.max = length(tsT70.imp),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Here, we observes most of the data lies outside the blue-dotted lines means the significance level.
Now, we will take the difference to make our data stationary.

It seems difference of 1 is sufficient to make the series stationary.

tsKI.sta <- diff(tsKI.imp)
acf(tsKI.sta, lag.max = length(tsKI.sta),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Hence, after differencing our series is now transformed to stationary time series.

#CREATING MODELS AND FORECASTING: USING EXPONENTIAL SMOOTHING AND ARIMA TECHNIQUES FOR TIME SERIES FORECASTING

#Visualising the stationary series

#plotting the transformed stationary series
plot(tsKI.sta, col = "red")

  1. TRAIN- TEST SET SPLIT
train7 <- window(tsKI.imp, end = c(2004, 3))
test7 <- window(tsKI.imp, start = c(2004, 4))

#Model 1 : Auto Exponential Smoothening Model

set.seed(301)
fit_auto_train_Var8 <- forecast(train7)
summary(fit_auto_train_Var8)
## 
## Forecast method: STL +  ETS(A,N,N)
## 
## Model Information:
## ETS(A,N,N) 
## 
## Call:
##  ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend) 
## 
##   Smoothing parameters:
##     alpha = 0.3354 
## 
##   Initial states:
##     l = 14.2489 
## 
##   sigma:  14.8835
## 
##      AIC     AICc      BIC 
## 28718.12 28718.13 28735.20 
## 
## Error measures:
##                      ME     RMSE      MAE MPE MAPE      MASE      ACF1
## Training set 0.01592438 14.87674 11.11423 Inf  Inf 0.5984462 0.2408974
## 
## Forecasts:
##           Point Forecast        Lo 80     Hi 80       Lo 95     Hi 95
## 2004.0082   18.880269119   -0.1937416  37.95428  -10.290913  48.05145
## 2004.0110   28.090116005    7.9719178  48.20831   -2.678013  58.85824
## 2004.0137   27.992483454    6.8816826  49.10328   -4.293700  60.27867
## 2004.0164   31.624741870    9.5659585  53.68353   -2.111256  65.36074
## 2004.0192    9.096112091  -13.8715595  32.06378  -26.029910  44.22213
## 2004.0219   15.296380701   -8.5455561  39.13832  -21.166715  51.75948
## 2004.0247   20.648317134   -4.0369409  45.33358  -17.104527  58.40116
## 2004.0274    8.434968649  -17.0657368  33.93567  -30.564995  47.43493
## 2004.0301   21.361635444   -4.9292373  47.65251  -18.846785  61.57006
## 2004.0329   11.016897487  -16.0410774  38.07487  -30.364704  52.39850
## 2004.0356   13.634269363  -14.1696516  41.43819  -28.888158  56.15670
## 2004.0384   20.632098920   -7.8982714  49.16247  -23.001337  64.26554
## 2004.0411   24.126739699   -5.1120367  53.36552  -20.590110  68.84359
## 2004.0438   30.134130746    0.2037105  60.06455  -15.640497  75.90876
## 2004.0466   11.517810221  -19.0886282  42.12425  -35.290698  58.32632
## 2004.0493   22.387615713   -8.8802286  53.65546  -25.432425  70.20766
## 2004.0521   17.013288253  -14.9022581  48.92883  -31.797328  65.82390
## 2004.0548    6.557020871  -25.9933418  39.10738  -43.224463  56.33850
## 2004.0575    2.741376067  -30.4316571  35.91441  -47.992400  53.47515
## 2004.0603   17.059782153  -16.7244470  50.84401  -34.608738  68.72830
## 2004.0630   10.029250521  -24.3553122  44.41381  -42.557400  62.61590
## 2004.0658   23.029083963  -11.9455092  58.00368  -30.459940  76.51811
## 2004.0685   15.985678246  -19.5691552  51.54051  -38.390747  70.36210
## 2004.0712    1.538199489  -34.5875558  37.66395  -53.711375  56.78777
## 2004.0740    5.807256382  -30.8805373  42.49505  -50.301882  61.91639
## 2004.0767   15.671677248  -21.5696737  52.91303  -41.284054  72.62741
## 2004.0795   23.767774695  -14.0190250  61.55457  -34.022149  81.55770
## 2004.0822   23.278159899  -15.0463264  61.60265  -35.334084  81.89040
## 2004.0849   27.079270172  -11.7754626  65.93400  -32.343916  86.50246
## 2004.0877   17.181044951  -22.1967949  56.55888  -43.042164  77.40425
## 2004.0904   18.056181698  -21.8379067  57.95027  -42.956561  79.06892
## 2004.0932   28.264400931  -12.1393403  68.66814  -33.527789  90.05659
## 2004.0959   11.362886107  -29.5441587  52.26993  -51.199040  73.92481
## 2004.0986   11.615294457  -29.7889364  53.01953  -51.707012  74.93760
## 2004.1014    7.139396851  -34.7561201  49.03491  -56.934267  71.21306
## 2004.1041    9.188190800  -33.1929177  51.56930  -55.628121  74.00450
## 2004.1068   12.833606304  -30.0275926  55.69481  -52.716941  78.38415
## 2004.1096   25.168002869  -18.1679681  68.50397  -41.108645  91.44465
## 2004.1123   14.887916867  -28.9176809  58.69351  -52.106964  81.88280
## 2004.1151   20.562243332  -23.7079996  64.83249  -47.143251  88.26774
## 2004.1178   26.142123528  -18.5879382  70.87219  -42.266603  94.55085
## 2004.1205   20.450762835  -24.7344387  65.63596  -48.654040  89.55557
## 2004.1233    0.216738388  -45.4190639  45.85254  -69.577198  70.01068
## 2004.1260    9.690306711  -36.3916905  55.77230  -60.786026  80.16664
## 2004.1288   16.795142737  -29.7287702  63.31906  -54.357042  87.94733
## 2004.1315   21.157622359  -25.8040481  68.11929  -50.664055  92.97930
## 2004.1342   26.046995187  -21.3483897  73.44238  -46.437991  98.53198
## 2004.1370   11.236546173  -36.5886200  59.06171  -61.905734  84.37883
## 2004.1397    5.522144366  -42.7289751  53.77326  -68.271575  79.31586
## 2004.1425   14.397208105  -34.2761372  63.07055  -60.042250  88.83667
## 2004.1452   21.080217937  -28.0117219  70.17216  -53.999425  96.15986
## 2004.1479   23.120652307  -26.3863429  72.62765  -52.593763  98.83507
## 2004.1507   33.447796921  -16.4708027  83.36640  -42.896113 109.79171
## 2004.1534   18.657029080  -31.6698087  68.98387  -58.311227  95.62529
## 2004.1562   21.856864632  -28.8749263  72.58866  -55.730714  99.44444
## 2004.1589   23.957806403  -27.1757308  75.09134  -54.244190 102.15980
## 2004.1616    3.181118701  -48.3510328  54.71327  -75.630506  81.99274
## 2004.1644   18.756933734  -33.1707723  70.68464  -60.659639  98.17351
## 2004.1671   27.941333934  -24.3789362  80.26160  -52.075614 107.95828
## 2004.1699   15.563827832  -37.1460828  68.27374  -65.049024  96.17668
## 2004.1726   24.900626317  -28.1960656  77.99732  -56.303756 106.10501
## 2004.1753   27.526303523  -25.9543725  81.00698  -54.265332 109.31794
## 2004.1781   20.273941434  -33.5879813  74.13586  -62.100761 102.64864
## 2004.1808   26.248734579  -27.9917552  80.48922  -56.704936 109.20241
## 2004.1836    3.257296699  -51.3591362  57.87373  -80.271329  86.78592
## 2004.1863   12.061090960  -42.9287150  67.05090  -72.038560  96.16074
## 2004.1890    8.170267190  -47.1903937  63.53093  -76.496557  92.83709
## 2004.1918    3.072079221  -52.6569687  58.80113  -82.158145  88.30230
## 2004.1945   11.653018862  -44.4419970  67.74803  -74.136905  97.44294
## 2004.1973   33.228386511  -23.2302250  89.68700  -53.117609 119.57438
## 2004.2000   32.140889334  -24.6789912  88.96077  -54.757619 119.03940
## 2004.2027   30.786462367  -26.3924047  87.96533  -56.661069 118.23399
## 2004.2055   31.110109544  -26.4255043  88.64572  -56.883018 119.10324
## 2004.2082   30.641157178  -27.2490049  88.53132  -57.894206 119.17652
## 2004.2110   25.329533742  -32.9130184  83.57209  -63.744763 114.40383
## 2004.2137   16.523344042  -42.0694789  75.11617  -73.086646 106.13333
## 2004.2164    9.586450044  -49.3545621  68.52746  -80.556049  99.72895
## 2004.2192   19.410752563  -39.8764039  78.69791  -71.261129 110.08263
## 2004.2219   23.876615298  -35.7546763  83.50791  -67.321575 115.07481
## 2004.2247   21.821187669  -38.1522644  81.79464  -69.900292 113.54267
## 2004.2274   18.978814332  -41.3348571  79.29249  -73.262986 111.22061
## 2004.2301   17.711127243  -42.9408552  78.36311  -75.048075 110.47033
## 2004.2329   21.437896663  -39.5505202  82.42631  -71.835838 114.71163
## 2004.2356   15.009968468  -46.3130370  76.33297  -78.775475 108.79541
## 2004.2384   15.524562989  -46.1312155  77.18034  -78.769813 109.81894
## 2004.2411   22.006507138  -39.9802578  83.99327  -72.794069 116.80708
## 2004.2438   35.635903579  -26.6800899  97.95190  -59.668184 130.93999
## 2004.2466   35.555339075  -27.0881527  98.19883  -60.249614 131.36029
## 2004.2493   31.097412158  -31.8718746  94.06670  -65.205801 127.40063
## 2004.2521   25.441565485  -37.8518393  88.73497  -71.357344 122.24047
## 2004.2548   27.923390201  -35.6924812  91.53926  -69.368689 125.21547
## 2004.2575   21.818113088  -42.1185987  85.75482  -75.964649 119.60088
## 2004.2603   21.534438272  -42.7215118  85.79039  -76.736557 119.80543
## 2004.2630   12.810555548  -51.7630547  77.38417  -85.946259 111.56737
## 2004.2658   16.354598160  -48.5351171  81.24431  -82.885657 115.59485
## 2004.2685   19.427540035  -45.7767479  84.63183  -80.293813 119.14889
## 2004.2712   11.308935208  -54.2084150  76.82629  -88.891205 111.50908
## 2004.2740   19.022002118  -46.8069216  84.85093  -81.654649 119.69865
## 2004.2767   22.620740603  -43.5182888  88.75977  -78.530176 123.77166
## 2004.2795   19.332448425  -47.1152394  85.78014  -82.290521 120.95542
## 2004.2822   22.258524969  -44.4963942  89.01344  -79.834314 124.35136
## 2004.2849   20.695403209  -46.3653397  87.75615  -81.865153 123.25596
## 2004.2877   25.110437113  -42.2547413  92.47562  -77.915713 128.13659
## 2004.2904   18.499860579  -49.1683836  86.16810  -84.989788 121.98951
## 2004.2932   27.318672138  -40.6512866  95.28863  -76.632409 131.26975
## 2004.2959   25.171313154  -43.0990267  93.44165  -79.239162 129.58179
## 2004.2986   28.934201197  -39.6352039  97.50361  -75.933654 133.80206
## 2004.3014   34.177254659  -34.6899170 103.04443  -71.145995 139.50050
## 2004.3041   35.553088310  -33.6105679 104.71674  -70.223596 141.32977
## 2004.3068   29.925706452  -39.5331688  99.38458  -76.302476 136.15389
## 2004.3096   29.774574870  -39.9782700  99.52742  -76.903196 136.45235
## 2004.3123   32.551928949  -37.4936518 102.59751  -74.573542 139.67740
## 2004.3151   23.771052765  -46.5660455  94.10815  -83.800256 131.34236
## 2004.3178   28.722123390  -41.9052892  99.34954  -79.293183 136.73743
## 2004.3205   27.549865731  -43.3666727  98.46640  -80.907620 136.00735
## 2004.3233   29.630800512  -41.5736898 100.83529  -79.267070 138.52867
## 2004.3260   32.205083273  -39.2861991 103.69637  -77.131398 141.54156
## 2004.3288   41.492567330  -30.2843611 113.26950  -68.280772 151.26591
## 2004.3315   20.751188140  -51.3102542  92.81263  -89.457278 130.95965
## 2004.3342   16.391298097  -55.9535392  88.73614  -94.250583 127.03318
## 2004.3370   26.704841424  -45.9222851  99.33197  -84.368763 137.77845
## 2004.3397   34.553941744  -38.3543809 107.46226  -76.949716 146.05760
## 2004.3425   28.944217190  -44.2442213 102.13266  -82.987840 140.87627
## 2004.3452   17.901484484  -55.5660018  91.36897  -94.457340 130.26031
## 2004.3479   22.485112991  -51.2603652  96.23059  -90.298863 135.26909
## 2004.3507   27.928351721  -46.0940744 101.95078  -85.279180 141.13588
## 2004.3534   31.728469752  -42.5698720 106.02681  -81.901038 145.35798
## 2004.3562   24.050082740  -50.5231537  98.62332  -89.999841 138.10001
## 2004.3589   17.332488140  -57.5146335  92.17961  -97.136306 131.80128
## 2004.3616   35.988474233  -39.1315339 111.10848  -78.897664 150.87461
## 2004.3644   33.311461768  -42.0804452 108.70337  -81.990510 148.61343
## 2004.3671   36.063841227  -39.5989875 111.72667  -79.652470 151.78015
## 2004.3699   37.097667261  -38.8351166 113.03045  -79.031504 153.22684
## 2004.3726   33.890052530  -42.3117301 110.09184  -82.650517 150.43062
## 2004.3753   33.822035091  -42.6478001 110.29187  -83.128486 150.77256
## 2004.3781   39.122967691  -37.6139837 115.85992  -78.236072 156.48201
## 2004.3808   31.887385312  -45.1157557 108.89053  -85.878756 149.65353
## 2004.3836   29.728702897  -47.5397106 106.99712  -88.443138 147.90054
## 2004.3863   37.576462720  -39.9563158 115.10924  -80.999690 156.15261
## 2004.3890   43.975799615  -33.8204455 121.77204  -75.003290 162.95489
## 2004.3918   42.640092152  -35.4187304 120.69891  -76.740575 162.02076
## 2004.3945   45.603734909  -32.7167847 123.92425  -74.177163 165.38463
## 2004.3973   39.861609121  -38.7197360 118.44295  -80.318187 160.04141
## 2004.4000   43.412981891  -35.4283259 122.25429  -77.164393 163.99036
## 2004.4027   43.252170442  -35.8482457 122.35259  -77.721477 164.22582
## 2004.4055   42.944751885  -36.4139265 122.30343  -78.423873 164.31338
## 2004.4082   41.492540116  -38.1235629 121.10864  -80.269782 163.25486
## 2004.4110   27.178755978  -52.6939419 107.05145  -94.975994 149.33351
## 2004.4137   34.293002690  -45.8354684 114.42147  -88.252919 156.83892
## 2004.4164   38.828027126  -41.5554033 119.21146  -84.107821 161.76388
## 2004.4192   39.497738061  -41.1398457 120.13532  -83.826804 162.82228
## 2004.4219   39.711596473  -41.1793420 120.60253  -84.000418 163.42361
## 2004.4247   41.821443892  -39.3220582 122.96495  -82.276834 165.91972
## 2004.4274   36.044075517  -45.3512066 117.43936  -88.439266 160.52742
## 2004.4301   38.806536470  -42.8397492 120.45282  -86.060682 163.67375
## 2004.4329   40.170682144  -41.7258378 122.06720  -85.079237 165.42060
## 2004.4356   42.061120990  -40.0848709 124.20711  -83.570332 167.69257
## 2004.4384   39.013398331  -43.3813102 121.40811  -86.998434 165.02523
## 2004.4411   38.717526259  -43.9251505 121.36020  -87.673541 165.10859
## 2004.4438   34.694809351  -48.1950937 117.58471  -92.074358 161.46398
## 2004.4466   37.662174380  -45.4742198 120.79857  -89.483968 164.80832
## 2004.4493   42.324875840  -41.0572809 125.70703  -85.197128 169.84688
## 2004.4521   42.198619564  -41.4285774 125.82582  -85.698141 170.09538
## 2004.4548   46.406600309  -37.4649210 130.27812  -81.863822 174.67702
## 2004.4575   44.969034997  -39.1461010 129.08417  -83.673964 173.61203
## 2004.4603   45.215619306  -39.1424278 129.57367  -83.798880 174.23012
## 2004.4630   42.968289598  -41.6319712 127.56855  -86.416644 172.35322
## 2004.4658   45.748242120  -39.0935408 130.59003  -84.006068 175.50255
## 2004.4685   48.471860603  -36.6107589 133.55448  -81.650777 178.59450
## 2004.4712   45.264923321  -40.0578530 130.58770  -85.225002 175.75485
## 2004.4740   40.707023148  -44.8552359 126.26928  -90.149160 171.56321
## 2004.4767   45.016585865  -40.7844875 130.81766  -86.204832 176.23800
## 2004.4795   43.779324149  -42.2599006 129.81855  -87.806315 175.36496
## 2004.4822   42.515374428  -43.7613444 128.79209  -89.433481 174.46423
## 2004.4849   47.167778390  -39.3457826 133.68134  -85.143295 179.47885
## 2004.4877   44.471918307  -42.2778381 131.22167  -88.200385 177.14422
## 2004.4904   41.854779220  -45.1305314 128.84009  -91.177773 174.88733
## 2004.4932   40.830191753  -46.3900368 128.05042  -92.561637 174.22202
## 2004.4959   42.637000372  -44.8175151 130.09152  -91.113139 176.38714
## 2004.4986   45.639852491  -42.0483240 133.32803  -88.467641 179.74735
## 2004.5014   45.747153980  -42.1740625 133.66837  -88.716743 180.21105
## 2004.5041   42.733209208  -45.4204312 130.88685  -92.086150 177.55257
## 2004.5068   41.054938693  -47.3305145 129.44039  -94.118947 176.22882
## 2004.5096   44.189449636  -44.4272099 132.80611  -91.338036 179.71694
## 2004.5123   41.386382277  -47.4608820 130.23365  -94.493783 177.26655
## 2004.5151   44.201712728  -44.8755592 133.27898  -92.030219 180.43364
## 2004.5178   44.651345122  -44.6553422 133.95803  -91.931447 181.23414
## 2004.5205   44.605942933  -44.9295718 134.14146  -92.326810 181.53870
## 2004.5233   48.497565271  -41.2661937 138.26132  -88.784257 185.77939
## 2004.5260   45.842838874  -44.1485853 135.83426  -91.787168 183.47285
## 2004.5288   39.559759545  -50.6587554 129.77827  -98.417552 177.53707
## 2004.5315   41.407385733  -49.0376498 131.85242  -96.916360 179.73113
## 2004.5342   39.560096805  -51.1108934 130.23109  -99.109216 178.22941
## 2004.5370   45.239076060  -45.6573072 136.13546  -93.774946 184.25310
## 2004.5397   47.601998215  -43.5192205 138.72322  -91.755880 186.95988
## 2004.5425   45.688780016  -45.6567208 137.03428  -94.012108 185.38967
## 2004.5452   41.090575961  -50.4786576 132.65981  -98.952482 181.13363
## 2004.5479   43.840658663  -47.9517623 135.63308  -96.543735 184.22505
## 2004.5507   42.029909754  -49.9851572 134.04498  -98.694991 182.75481
## 2004.5534   39.946400242  -52.2907754 132.18358 -101.118187 181.01099
## 2004.5562   42.785532646  -49.6732181 135.24428  -98.617924 184.18899
## 2004.5589   45.742095564  -46.9377005 138.42189  -95.999421 187.48361
## 2004.5616   46.479949379  -46.4203661 139.38026  -95.598822 188.55872
## 2004.5644   43.312211479  -49.8081011 136.43252  -99.103017 185.72744
## 2004.5671   46.748265694  -46.5915256 140.08806  -96.002626 189.49916
## 2004.5699   46.623145196  -46.9356099 140.18190  -96.462623 189.70891
## 2004.5726   45.026144543  -48.7510630 138.80335  -98.393718 188.44601
## 2004.5753   49.308369024  -44.6867834 143.30352  -94.444811 193.06155
## 2004.5781   47.177318225  -47.0352748 141.38991  -96.908409 191.26305
## 2004.5808   42.916705440  -51.5128276 137.34624 -101.500803 187.33421
## 2004.5836   43.098293661  -51.5476820 137.74427 -101.650235 187.84682
## 2004.5863   48.683641685  -46.1782829 143.54557  -96.395152 193.76244
## 2004.5890   41.378801238  -53.6985817 136.45618 -104.029508 186.78711
## 2004.5918   44.955258976  -50.3370952 140.24761 -100.781820 190.69234
## 2004.5945   43.100584354  -52.4062571 138.60743 -102.964525 189.16569
## 2004.5973   46.191362991  -49.5294852 141.91221 -100.201041 192.58377
## 2004.6000   44.257539218  -51.6768383 140.19192 -102.461430 190.97651
## 2004.6027   45.428418126  -50.7190146 141.57585 -101.616391 192.47323
## 2004.6055   38.779893595  -57.5801231 135.13991 -108.590035 186.14982
## 2004.6082   45.010218435  -51.5619144 141.58235 -102.684113 192.70455
## 2004.6110   41.285909480  -55.4978746 138.06969 -106.732115 189.30393
## 2004.6137   40.936901892  -56.0580715 137.93188 -107.404109 189.27791
## 2004.6164   42.732183032  -54.4735209 139.93789 -105.931112 191.39548
## 2004.6192   45.859841518  -51.5561371 143.27582 -103.125041 194.84472
## 2004.6219   34.320962963  -63.3048374 131.94676 -114.984814 183.62674
## 2004.6247   35.642283249  -62.1928889 133.47746 -113.983701 185.26827
## 2004.6274   43.788809439  -54.2552874 141.83291 -106.156697 193.73432
## 2004.6301   44.793740485  -53.4588368 143.04632 -105.470609 195.05809
## 2004.6329   49.288972020  -49.1716443 147.74959 -101.293546 199.87149
## 2004.6356   50.716090367  -47.9521263 149.38431 -100.183925 201.61611
## 2004.6384   47.224855576  -51.6505255 146.10024 -103.991991 198.44170
## 2004.6411   46.456338311  -52.6257741 145.53845 -105.076676 197.98935
## 2004.6438   40.166301714  -59.1221116 139.45472 -111.682223 192.01483
## 2004.6466   42.185966022  -57.3083204 141.68025 -109.977414 194.34935
## 2004.6493   42.209535683  -57.4901987 141.90927 -110.268050 194.68712
## 2004.6521   35.276997548  -64.6277624 135.18176 -117.514148 188.06814
## 2004.6548   38.997354432  -61.1120111 139.10672 -114.106708 192.10142
## 2004.6575   37.603466932  -62.7100869 137.91702 -115.812875 191.01981
## 2004.6603   30.207151695  -70.3101756 130.72448 -123.520835 183.93514
## 2004.6630   38.083964152  -62.6367244 138.80465 -115.955036 192.12296
## 2004.6658   47.178725172  -53.7449149 148.10237 -107.170663 201.52811
## 2004.6685   32.241154504  -68.8850297 133.36734 -122.417998 186.90031
## 2004.6712   33.585687527  -67.7426360 134.91401 -121.382610 188.55399
## 2004.6740   35.348971078  -66.1810893 136.87903 -119.927857 190.62580
## 2004.6767   28.895892713  -72.8355045 130.62729 -126.688853 184.48064
## 2004.6795   32.704518445  -69.2278179 134.63685 -123.187537 188.59657
## 2004.6822   38.271817692  -63.8610624 140.40470 -117.926943 194.47058
## 2004.6849   32.153293441  -70.1797375 134.48632 -124.351572 188.65816
## 2004.6877   32.204105051  -70.3286859 134.73690 -124.606267 189.01448
## 2004.6904   31.611480098  -71.1206826 134.34364 -125.503804 188.72676
## 2004.6932   24.213046428  -78.7181017 127.14419 -133.206560 181.63265
## 2004.6959   16.710352058  -86.4193976 119.84010 -141.012989 174.43369
## 2004.6986   28.063565923  -75.2644036 131.39154 -129.962927 186.09006
## 2004.7014   19.911454185  -83.6143556 123.43726 -138.417609 178.24052
## 2004.7041   21.487171019  -82.2361017 125.21044 -137.143886 180.11823
## 2004.7068   27.696886174  -76.2234743 131.61725 -131.235590 186.62936
## 2004.7096   31.816808737  -72.3002664 135.93388 -127.416517 191.05013
## 2004.7123   25.896362044  -78.4170568 130.20978 -133.637245 185.42997
## 2004.7151   18.590424341  -85.9189693 123.09982 -141.242900 178.42375
## 2004.7178    5.793477291  -98.9115244 110.49848 -154.339004 165.92596
## 2004.7205   12.524404879  -92.3758401 117.42465 -147.906675 172.95549
## 2004.7233   24.209971195  -80.8851543 129.30510 -136.519153 184.93910
## 2004.7260   33.806593656  -71.4830517 139.09624 -127.220023 194.83321
## 2004.7288   29.925832856  -75.5579736 135.40964 -131.397728 191.24939
## 2004.7315   23.399205628  -82.2784053 129.07682 -138.220753 185.01916
## 2004.7342   29.103283202  -76.7677773 134.97434 -132.812531 191.01910
## 2004.7370   17.404858666  -88.6592987 123.46902 -144.806272 179.61599
## 2004.7397   21.505138907  -84.7517643 127.76204 -141.000771 184.01105
## 2004.7425   35.756576275  -70.6927239 142.20588 -127.043579 198.55673
## 2004.7452   31.339167345  -75.3021826 137.98052 -131.754703 194.43304
## 2004.7479   28.088568082  -78.7444864 134.92162 -135.298489 191.47563
## 2004.7507   21.794251503  -85.2301641 128.81867 -141.885467 185.47397
## 2004.7534   20.460183792  -86.7552514 127.67562 -143.511674 184.43204
## 2004.7562   22.781944749  -84.6241704 130.18806 -141.481533 187.04542
## 2004.7589   24.238186315  -83.3582708 131.83464 -140.316395 188.79277
## 2004.7616   29.278887962  -78.5075750 137.06535 -135.566282 194.12406
## 2004.7644   23.788320851  -84.1878136 131.76446 -141.346926 188.92357
## 2004.7671   20.985190714  -87.1802826 129.15066 -144.439625 186.41001
## 2004.7699    8.816776303  -99.5377051 117.17126 -156.897103 174.53066
## 2004.7726   14.985407920  -93.5577524 123.52857 -151.017031 180.98785
## 2004.7753   25.567823645  -83.1636882 134.29934 -140.722674 191.85832
## 2004.7781   18.990750478  -89.9287871 127.91029 -147.587307 185.56881
## 2004.7808   21.234797135  -87.8724423 130.34204 -145.630326 188.09992
## 2004.7836   17.992357761  -91.3022611 127.28698 -149.159337 185.14405
## 2004.7863   23.189046005  -86.2926315 132.67072 -144.248731 190.62682
## 2004.7890    7.451932932 -102.2164843 117.12035 -160.271437 175.17530
## 2004.7918    2.653757020 -107.2010824 112.50860 -165.354722 170.66224
## 2004.7945    6.007289643 -104.0336562 116.04824 -162.285814 174.30039
## 2004.7973   16.180030526  -94.0467075 126.40677 -152.397218 184.75728
## 2004.8000   20.120778622  -90.2914389 130.53300 -148.740136 188.98169
## 2004.8027   20.734260823  -89.8631252 131.33165 -148.409845 189.87837
## 2004.8055   15.708147188  -95.0740978 126.49039 -153.718676 185.13497
## 2004.8082   15.368723119  -95.5980729 126.33552 -154.340346 185.07779
## 2004.8110   24.070679693  -87.0803610 135.22172 -145.920168 194.06153
## 2004.8137   21.659312035  -89.6756683 132.99429 -148.612847 191.93147
## 2004.8164   13.556988671  -97.9616280 125.07561 -156.996018 184.11000
## 2004.8192   20.110699154  -91.5912519 131.81265 -150.722693 190.94409
## 2004.8219    9.234050999 -102.6509341 121.11904 -161.879267 180.34737
## 2004.8247   13.105201055  -98.9625191 125.17292 -158.287587 184.49799
## 2004.8274   14.277208773  -97.9729490 126.52737 -157.394593 185.94901
## 2004.8301   17.026515479  -95.4057838 129.45881 -154.923848 188.97688
## 2004.8329   21.084056041  -91.5300902 133.69820 -151.144418 193.31253
## 2004.8356    5.349270103 -107.4464300 118.14497 -167.156867 177.85541
## 2004.8384   -2.109429416 -115.0863915 110.86753 -174.892783 170.67392
## 2004.8411   23.061021235  -90.0969126 136.21896 -149.999104 196.12115
## 2004.8438   28.751757721  -84.5868588 142.09037 -144.584698 202.08821
## 2004.8466   15.765399564  -97.7536121 129.28441 -157.846947 189.37775
## 2004.8493   10.873204646 -102.8259160 124.57233 -163.014595 184.76100
## 2004.8521   -8.822743830 -122.7016885 105.05620 -182.985560 165.34007
## 2004.8548    1.454257606 -112.6042276 115.51274 -172.983142 175.89166
## 2004.8575   10.643375125 -103.5943685 124.88112 -164.068177 185.35493
## 2004.8603   20.132413449  -94.2843077 134.54913 -154.852861 195.11769
## 2004.8630   30.246725480  -84.3486937 144.84214 -145.011844 205.50530
## 2004.8658   10.456461603 -104.3173773 125.23030 -165.074978 185.98790
## 2004.8685   10.121403216 -104.8305786 125.07339 -165.682482 185.92529
## 2004.8712    5.399322151 -109.7305269 120.52917 -170.676587 181.47523
## 2004.8740   13.395743794 -101.9116981 128.70319 -162.951771 189.74326
## 2004.8767   21.568312328  -93.9164493 137.05307 -155.050389 198.18701
## 2004.8795   16.667305525  -98.9945040 132.32912 -160.222167 193.55678
## 2004.8822    7.104670768 -108.7339160 122.94326 -170.055160 184.26450
## 2004.8849   11.588398610 -104.4266961 127.60349 -165.841377 189.01817
## 2004.8877   20.951828608  -95.2395059 137.14316 -156.747483 198.65114
## 2004.8904   19.720259780  -96.6470476 136.08757 -158.248179 197.68870
## 2004.8932   13.480305592 -103.0627090 130.02332 -164.756854 191.71747
## 2004.8959    6.037908877 -110.6805483 122.75637 -172.467567 184.54339
## 2004.8986    4.982663514 -111.9109730 121.87630 -173.790726 183.75605
## 2004.9014   -0.007340235 -117.0758940 117.06121 -179.048243 179.03356
## 2004.9041   17.584333165  -99.6588769 134.82754 -161.723683 196.89235
## 2004.9068   14.042840404 -103.3747661 131.46045 -165.531892 193.61757
## 2004.9096    0.193688408 -117.3980559 117.78543 -179.647365 180.03474
## 2004.9123   21.487598717  -96.2780259 139.25322 -158.619382 201.59458
## 2004.9151   35.502002978  -82.4372456 153.44125 -144.870512 215.87452
## 2004.9178   15.264535242 -102.8480820 133.37715 -165.373125 195.90220
## 2004.9205   13.287041452 -104.9986905 131.57277 -167.615375 194.18946
## 2004.9233   15.019356383 -103.4392372 133.47795 -166.147429 196.18614
## 2004.9260   10.462744577 -108.1684587 129.09395 -170.968024 191.89351
## 2004.9288    1.258169717 -117.5453926 120.06173 -180.436200 182.95254
## 2004.9315    4.015259069 -114.9604125 122.99093 -177.942329 185.97285
## 2004.9342  -17.337199570 -136.4847318 101.81033 -199.557625 164.88323
## 2004.9370    8.805058380 -110.5140870 128.12420 -173.677827 191.28794
## 2004.9397   22.703273752  -96.7872383 142.19379 -160.041694 205.44824
## 2004.9425   19.439372424 -100.2222609 139.10101 -163.567303 202.44605
## 2004.9452    8.365092064 -111.4674181 128.19760 -174.902917 191.63310
## 2004.9479    9.364797592 -110.6383462 129.36794 -174.164173 192.89377
## 2004.9507   17.513970255 -102.6595648 137.68751 -166.275591 201.30353
## 2004.9534   13.649091017 -106.6945941 133.99278 -170.400693 197.69887
## 2004.9562   16.359238260 -104.1543566 136.87283 -167.950400 200.66888
## 2004.9589    8.315842121 -112.3674233 128.99911 -176.253285 192.88497
## 2004.9616   11.913976055 -108.9387218 132.76667 -172.914276 196.74223
## 2004.9644   17.207938518 -103.8139545 138.22983 -167.879075 202.29495
## 2004.9671   15.218028182 -105.9728238 136.40888 -170.127386 200.56344
## 2004.9699    2.446018792 -118.9135569 123.80559 -183.157436 188.04947
## 2004.9726    6.112418176 -115.4156470 127.64048 -179.748719 191.97356
## 2004.9753    5.659208178 -116.0371132 127.35553 -180.459254 191.77767
## 2004.9781    3.537272747 -118.3270726 125.40162 -182.838160 189.91271
## 2004.9808   10.089589069 -111.9425488 132.12173 -176.542460 196.72164
## 2004.9836   12.231175991 -109.9685240 134.43088 -174.657138 199.11949
## 2004.9863   16.863428571 -105.5036042 139.23046 -170.280798 204.00766
## 2004.9890   15.401866603 -107.1322703 137.93600 -171.997924 202.80166
## 2004.9918   -1.723815351 -124.4248289 120.97720 -189.378822 185.93119
## 2004.9945   11.778424288 -111.0892392 134.64609 -176.131451 199.68830
## 2004.9973   13.395149732 -109.6389380 136.42924 -174.769250 201.55955
## 2005.0000   11.403125811 -111.7971614 134.60341 -177.015454 199.82171
## 2005.0027   32.225373152  -91.1408896 155.59164 -156.447044 220.89779
## 2005.0055   16.897052145 -106.6349631 140.42907 -172.028862 205.82297
## 2005.0082   18.880269119 -104.8172766 142.57781 -170.298802 208.05934
## 2005.0110   28.090116005  -95.7727389 151.95297 -161.341774 217.52201
## 2005.0137   27.992483454  -96.0354603 152.02043 -161.691888 217.67685
## 2005.0164   31.624741870  -92.5680713 155.81756 -158.311775 221.56126
## 2005.0192    9.096112091 -115.2613519 133.45358 -181.092217 199.28444
## 2005.0219   15.296380701 -109.2255165 139.81828 -175.143427 205.73619
## 2005.0247   20.648317134 -104.0377963 145.33443 -170.042638 211.33927
## 2005.0274    8.434968649 -116.4151451 133.28508 -182.506803 199.37674
## 2005.0301   21.361635444 -103.6522634 146.37553 -169.830624 212.55389
## 2005.0329   11.016897487 -114.1605722 136.19437 -180.425522 202.45932
## 2005.0356   13.634269363 -111.7065578 138.97510 -178.057984 205.32652
## 2005.0384   20.632098920 -104.8718730 146.13607 -171.309662 212.57386
## 2005.0411   24.126739699 -101.5401651 149.79364 -168.064206 216.31769
## 2005.0438   30.134130746  -95.6954961 155.96376 -162.305677 222.57394
## 2005.0466   11.517810221 -114.4743284 137.50995 -181.170538 204.20616
## 2005.0493   22.387615713 -103.7668254 148.54206 -170.548953 215.32418
## 2005.0521   17.013288253 -109.3032468 143.32982 -176.171181 210.19776
## 2005.0548    6.557020871 -119.9214004 133.03544 -186.875032 199.98907
## 2005.0575    2.741376067 -123.8987244 129.38148 -190.937944 196.42070
## 2005.0603   17.059782153 -109.7417914 143.86136 -176.866490 210.98605
## 2005.0630   10.029250521 -116.9335908 136.99209 -184.143659 204.20216
## 2005.0658   23.029083963 -104.0948205 150.15299 -171.390151 217.44832
## 2005.0685   15.985678246 -111.2990856 143.27044 -178.679569 210.65093
## 2005.0712    1.538199489 -125.9072207 128.98362 -193.372751 196.44915
## 2005.0740    5.807256382 -121.7986178 133.41313 -189.349087 200.96360
## 2005.0767   15.671677248 -112.0944495 143.43780 -179.729751 211.07311
## 2005.0795   23.767774695 -104.1584039 151.69395 -171.878432 219.41398
## 2005.0822   23.278159899 -104.8078705 151.36419 -172.612519 219.16884
## 2005.0849   27.079270172 -101.1664127 155.32495 -169.055576 223.21412
## 2005.0877   17.181044951 -111.2240920 145.58618 -179.197666 213.55976
## 2005.0904   18.056181698 -110.5082116 146.62057 -178.566090 214.67845
## 2005.0932   28.264400931 -100.4590516 156.98785 -168.601131 225.12993
## 2005.0959   11.362886107 -117.5194294 140.24520 -185.745606 208.47138
## 2005.0986   11.615294457 -117.4256884 140.65628 -185.735858 208.96645
## 2005.1014    7.139396851 -122.0600585 136.33885 -190.454119 204.73291
## 2005.1041    9.188190800 -120.1695429 138.54592 -188.647391 207.02377
## 2005.1068   12.833606304 -116.6822124 142.34943 -185.243745 210.91096
## 2005.1096   25.168002869 -104.5057081 154.84171 -173.150824 223.48683
## 2005.1123   14.887916867 -114.9434943 144.71933 -183.672092 213.44793
## 2005.1151   20.562243332 -109.4266767 150.55116 -178.238654 219.36314
## 2005.1178   26.142123528 -104.0041148 156.28836 -172.899372 225.18362
## 2005.1205   20.450762835 -109.8526038 150.75413 -178.831039 219.73257
## 2005.1233    0.216738388 -130.2435674 130.67704 -199.305082 199.73856
## 2005.1260    9.690306711 -120.9267496 140.30736 -190.071242 209.45186
## 2005.1288   16.795142737 -113.9784762 147.56876 -183.205848 216.79613
## 2005.1315   21.157622359 -109.7723720 152.08762 -179.082524 221.39777
## 2005.1342   26.046995187 -105.0391880 157.13318 -174.432022 226.52601
## 2005.1370   11.236546173 -120.0056400 142.47873 -189.481057 211.95415
## 2005.1397    5.522144366 -125.8758596 136.92015 -195.433761 206.47805
## 2005.1425   14.397208105 -117.1564291 145.95085 -186.796718 215.59113
## 2005.1452   21.080217937 -110.6288686 152.78930 -180.351447 222.51188
## 2005.1479   23.120652307 -108.7437003 154.98500 -178.548472 224.78978
## 2005.1507   33.447796921  -98.5716392 165.46723 -168.458507 235.35410
## 2005.1534   18.657029080 -113.5173085 150.83137 -183.486176 220.80023
## 2005.1562   21.856864632 -110.4721932 154.18592 -180.522965 224.23669
## 2005.1589   23.957806403 -108.5257909 156.44140 -178.658371 226.57398
## 2005.1616    3.181118701 -129.4568380 135.81908 -199.671131 206.03337
## 2005.1644   18.756933734 -114.0352030 151.54907 -184.331114 221.84498
## 2005.1671   27.941333934 -105.0048040 160.88747 -175.382238 231.26491
## 2005.1699   15.563827832 -117.5361331 148.66379 -187.994996 219.12265
## 2005.1726   24.900626317 -108.3529801 158.15423 -178.893178 228.69443
## 2005.1753   27.526303523 -105.8807714 160.93338 -176.502211 231.55482
## 2005.1781   20.273941434 -113.2864256 153.83431 -183.989013 224.53690
## 2005.1808   26.248734579 -107.4647489 159.96222 -178.248391 230.74586
## 2005.1836    3.257296699 -130.6091280 137.12372 -201.473733 207.98833
## 2005.1863   12.061090960 -121.9581005 146.08028 -192.903575 217.02576
## 2005.1890    8.170267190 -126.0015171 142.34205 -197.027769 213.36830
## 2005.1918    3.072079221 -131.2521245 137.39628 -202.359063 208.50322
## 2005.1945   11.653018862 -122.8234316 146.12947 -194.010964 217.31700
## 2005.1973   33.228386511 -101.4001385 167.85691 -172.668175 239.12495
## 2005.2000   32.140889334 -102.6395386 166.92132 -173.987987 238.26977
## 2005.2027   30.786462367 -104.1456975 165.71862 -175.574468 237.14739
## 2005.2055   31.110109544 -103.9736118 166.19383 -175.482614 237.70283
## 2005.2082   30.641157178 -104.5939558 165.87627 -176.183100 237.46541
## 2005.2110   25.329533742 -110.0568016 160.71587 -181.725998 232.38507
## 2005.2137   16.523344042 -119.0140450 152.06073 -190.763205 223.80989
## 2005.2164    9.586450044 -126.1018244 145.27472 -197.930858 217.10376
## 2005.2192   19.410752563 -116.4282398 155.24974 -188.337059 227.15856
## 2005.2219   23.876615298 -112.1129279 159.86616 -184.101444 231.85467
## 2005.2247   21.821187669 -114.3187399 157.96112 -186.386864 230.02924
## 2005.2274   18.978814332 -117.3113316 155.26896 -189.458977 227.41661
## 2005.2301   17.711127243 -118.7290717 154.15133 -190.956150 226.37840
## 2005.2329   21.437896663 -115.1521905 158.02798 -187.458615 230.33441
## 2005.2356   15.009968468 -121.7298426 151.74978 -194.115526 224.13546
## 2005.2384   15.524562989 -121.3648082 152.41393 -193.829664 224.87879
## 2005.2411   22.006507138 -115.0322609 159.04528 -187.576203 231.58922
## 2005.2438   35.635903579 -101.5520987 172.82391 -174.175040 245.44685
## 2005.2466   35.555339075 -101.7817352 172.89241 -174.483591 245.59427
## 2005.2493   31.097412158 -106.3885726 168.58340 -179.169257 241.36408
## 2005.2521   25.441565485 -112.1931685 163.07630 -185.052596 235.93573
## 2005.2548   27.923390201 -109.8599325 165.70671 -182.798018 238.64480
## 2005.2575   21.818113088 -116.1136383 159.74986 -189.130297 232.76652
## 2005.2603   21.534438272 -116.5455822 159.61446 -189.640730 232.70961
## 2005.2630   12.810555548 -125.4175750 151.03869 -198.591127 224.21224
## 2005.2658   16.354598160 -122.0214839 154.73068 -195.273357 227.98255
## 2005.2685   19.427540035 -119.0963355 157.95142 -192.426446 231.28153
## 2005.2712   11.308935208 -127.3625763 149.98045 -200.770840 223.38871
## 2005.2740   19.022002118 -119.7969884 157.84099 -193.283323 231.32733
## 2005.2767   22.620740603 -116.3455723 161.58705 -189.909895 235.15138
## 2005.2795   19.332448425 -119.7810309 158.44593 -193.423259 232.08816
## 2005.2822   22.258524969 -117.0019653 161.51902 -190.722016 235.23907
## 2005.2849   20.695403209 -118.7119430 160.10275 -192.509734 233.90054
## 2005.2877   25.110437113 -114.4436104 164.66448 -188.319061 238.53993
## 2005.2904   18.499860579 -121.2007343 158.20046 -195.153762 232.15348
## 2005.2932   27.318672138 -112.5283164 167.16566 -186.558840 241.19618
## 2005.2959   25.171313154 -114.8219161 165.16454 -188.929855 239.27248
## 2005.2986   28.934201197 -111.2051160 169.07352 -185.390389 243.25879
## 2005.3014   34.177254659 -106.1079985 174.46251 -180.370526 248.72503
## 2005.3041   35.553088310 -104.8779491 175.98413 -179.217650 250.32383
## 2005.3068   29.925706452 -110.6509640 170.50238 -185.067758 244.91917
## 2005.3096   29.774574870 -110.9475779 170.49673 -185.441386 244.99054
## 2005.3123   32.551928949 -108.3155560 173.41941 -182.886298 247.99016
## 2005.3151   23.771052765 -117.2416145 164.78372 -191.889211 239.43132
## 2005.3178   28.722123390 -112.4355768 169.87982 -187.159950 244.60420
## 2005.3205   27.549865731 -113.7527186 168.85245 -188.553788 243.65352
## 2005.3233   29.630800512 -111.8165196 171.07812 -186.694208 245.95581
## 2005.3260   32.205083273 -109.3868246 173.79699 -184.341053 248.75122
## 2005.3288   41.492567330 -100.2437808 183.22892 -175.274471 258.25961
## 2005.3315   20.751188140 -121.1294532 162.63183 -196.236528 237.73890
## 2005.3342   16.391298097 -125.6334899 158.41609 -200.816871 233.59947
## 2005.3370   26.704841424 -115.4639471 168.87363 -190.723558 244.13324
## 2005.3397   34.553941744 -107.7587015 176.86659 -183.094464 252.20235
## 2005.3425   28.944217190 -113.5121356 171.40057 -188.923973 246.81241
## 2005.3452   17.901484484 -124.6984330 160.50140 -200.186269 235.98924
## 2005.3479   22.485112991 -120.2582248 165.22845 -195.821983 240.79221
## 2005.3507   27.928351721 -114.9582624 170.81497 -190.597867 246.45457
## 2005.3534   31.728469752 -111.3012772 174.75822 -187.016652 250.47359
## 2005.3562   24.050082740 -119.1226539 167.22282 -194.913723 243.01389
## 2005.3589   17.332488140 -125.9830956 160.64807 -201.849783 236.51476
## 2005.3616   35.988474233 -107.4698143 179.44676 -183.412045 255.38899
## 2005.3644   33.311461768 -110.2893898 176.91231 -186.307089 252.93001
## 2005.3671   36.063841227 -107.6794319 179.80711 -183.772524 255.90021
## 2005.3699   37.097667261 -106.7878865 180.98322 -182.956298 257.15163
## 2005.3726   33.890052530 -110.1376413 177.91775 -186.381297 254.16140
## 2005.3753   33.822035091 -110.3476587 177.99173 -186.666485 254.31055
## 2005.3781   39.122967691 -105.1885863 183.43452 -181.582508 259.82844
## 2005.3808   31.887385312 -112.5658896 176.34066 -189.034834 252.80960
## 2005.3836   29.728702897 -114.8661540 174.32356 -191.410048 250.86745
## 2005.3863   37.576462720 -107.1598377 182.31276 -183.778607 258.93153
## 2005.3890   43.975799615 -100.9018063 188.85341 -177.595378 265.54698
## 2005.3918   42.640092152 -102.3786814 187.65887 -179.146983 264.42717
## 2005.3945   45.603734909  -99.5560691 190.76354 -176.399028 267.60650
## 2005.3973   39.861609121 -105.4390885 185.16231 -182.356632 262.07985
## 2005.4000   43.412981891 -102.0284728 188.85444 -179.020528 265.84649
## 2005.4027   43.252170442 -102.3299052 188.83425 -179.396401 265.90074
## 2005.4055   42.944751885 -102.7778090 188.66731 -179.918673 265.80818
## 2005.4082   41.492540116 -104.3703708 187.35545 -181.585532 264.57061
## 2005.4110   27.178755978 -118.8243700 173.18188 -196.113756 250.47127
## 2005.4137   34.293002690 -111.8502038 180.43621 -189.213744 257.79975
## 2005.4164   38.828027126 -107.4551258 185.11118 -184.892749 262.54880
## 2005.4192   39.497738061 -106.9252275 185.92070 -184.436863 263.43234
## 2005.4219   39.711596473 -106.8510483 186.27424 -184.436626 263.85982
## 2005.4247   41.821443892 -104.8807472 188.52363 -182.540196 266.18308
## 2005.4274   36.044075517 -110.7975292 182.88568 -188.530779 260.61893
## 2005.4301   38.806536470 -108.1743497 185.78742 -185.981331 263.59440
## 2005.4329   40.170682144 -106.9493535 187.29072 -184.829996 265.17136
## 2005.4356   42.061120990 -105.1979328 189.32017 -183.152167 267.27441
## 2005.4384   39.013398331 -108.3845424 186.41134 -186.412299 264.43910
## 2005.4411   38.717526259 -108.8191707 186.25422 -186.920380 264.35543
## 2005.4438   34.694809351 -112.9805134 182.37013 -191.155107 260.54473
## 2005.4466   37.662174380 -110.1516442 185.47599 -188.399553 263.72390
## 2005.4493   42.324875840 -105.6273089 190.27706 -183.948465 268.59822
## 2005.4521   42.198619564 -105.8918021 190.28904 -184.286136 268.68338
## 2005.4548   46.406600309 -101.8219294 194.63513 -180.289373 273.10257
## 2005.4575   44.969034997 -103.3974741 193.33554 -181.937960 271.87603
## 2005.4603   45.215619306 -103.2887410 193.71998 -181.902201 272.33344
## 2005.4630   42.968289598 -105.6737941 191.61037 -184.360160 270.29674
## 2005.4658   45.748242120 -103.0314375 194.52792 -181.790642 273.28713
## 2005.4685   48.471860603 -100.4452878 197.38901 -179.277264 276.22099
## 2005.4712   45.264923321 -103.7895670 194.31941 -182.694248 273.22409
## 2005.4740   40.707023148 -108.4846827 189.89873 -187.462001 268.87605
## 2005.4767   45.016585865 -104.3122095 194.34538 -183.362099 273.39527
## 2005.4795   43.779324149 -105.6864349 193.24508 -184.808828 272.36748
## 2005.4822   42.515374428 -107.0872229 192.11797 -186.282054 271.31280
## 2005.4849   47.167778390 -102.5715323 196.90709 -181.838735 276.17429
## 2005.4877   44.471918307 -105.4039809 194.34782 -184.743489 273.68733
## 2005.4904   41.854779220 -108.1575842 191.86714 -187.569333 271.27889
## 2005.4932   40.830191753 -109.3185119 190.97890 -188.802434 270.46282
## 2005.4959   42.637000372 -107.6479197 192.92192 -187.203951 272.47795
## 2005.4986   45.639852491 -104.7811607 196.06087 -184.409235 275.68894
## 2005.5014   45.747153980 -104.8098294 196.30414 -184.509882 276.00419
## 2005.5041   42.733209208 -107.9596216 193.42604 -187.731588 273.19801
## 2005.5068   41.054938693 -109.7736172 191.88349 -189.617432 271.72731
## 2005.5096   44.189449636 -106.7747093 195.15361 -186.690308 275.06921
## 2005.5123   41.386382277 -109.7132580 192.48602 -189.700576 272.47334
## 2005.5151   44.201712728 -107.0332875 195.43671 -187.092261 275.49569
## 2005.5178   44.651345122 -106.7188941 196.02158 -186.849458 276.15215
## 2005.5205   44.605942933 -106.8994145 196.11130 -187.101506 276.31339
## 2005.5233   48.497565271 -103.1427900 200.13792 -183.416345 280.41148
## 2005.5260   45.842838874 -105.9323941 197.61807 -186.277349 277.96303
## 2005.5288   39.559759545 -112.3502315 191.46975 -192.766523 271.88604
## 2005.5315   41.407385733 -110.6372438 193.45202 -191.124809 273.93958
## 2005.5342   39.560096805 -112.6190522 191.73925 -193.177828 272.29802
## 2005.5370   45.239076060 -107.0744736 197.55263 -187.704396 278.18255
## 2005.5397   47.601998215 -104.8458336 200.04983 -185.546841 280.75084
## 2005.5425   45.688780016 -106.8932158 198.27078 -187.665245 279.04281
## 2005.5452   41.090575961 -111.6254659 193.80662 -192.468455 274.64961
## 2005.5479   43.840658663 -109.0093118 196.69063 -189.923198 277.60452
## 2005.5507   42.029909754 -110.9538720 195.01369 -191.938594 275.99841
## 2005.5534   39.946400242 -113.1710759 193.06388 -194.226572 274.11937
## 2005.5562   42.785532646 -110.4655212 196.03659 -191.591729 277.16279
## 2005.5589   45.742095564 -107.6424197 199.12661 -188.839277 280.32347
## 2005.5616   46.479949379 -107.0379113 199.99781 -188.305358 281.26526
## 2005.5644   43.312211479 -110.3388788 196.96330 -191.676853 278.30128
## 2005.5671   46.748265694 -107.0359388 200.53247 -188.444379 281.94091
## 2005.5699   46.623145196 -107.2940584 200.54035 -188.772904 282.01919
## 2005.5726   45.026144543 -109.0239434 199.07623 -190.573134 280.62542
## 2005.5753   49.308369024 -104.8744886 203.49123 -186.493963 285.11070
## 2005.5781   47.177318225 -107.1381950 201.49283 -188.827893 283.18253
## 2005.5808   42.916705440 -111.5313493 197.36476 -193.291211 279.12462
## 2005.5836   43.098293661 -111.4821891 197.67878 -193.312154 279.50874
## 2005.5863   48.683641685 -106.0291556 203.39644 -187.929163 285.29645
## 2005.5890   41.378801238 -113.4661976 196.22380 -195.436189 278.19379
## 2005.5918   44.955258976 -110.0218286 199.93235 -192.061743 281.97226
## 2005.5945   43.100584354 -112.0084795 198.20965 -194.118258 280.31943
## 2005.5973   46.191362991 -109.0495649 201.43229 -191.229148 283.61187
## 2005.6000   44.257539218 -111.1151409 199.63022 -193.364469 281.87955
## 2005.6027   45.428418126 -110.0759025 200.93274 -192.394917 283.25175
## 2005.6055   38.779893595 -116.8559562 194.41574 -199.244598 276.80439
## 2005.6082   45.010218435 -110.7570495 200.77749 -193.215260 283.23570
## 2005.6110   41.285909480 -114.6126658 197.18448 -197.140386 279.71221
## 2005.6137   40.936901892 -115.0928702 196.96667 -197.690042 279.56385
## 2005.6164   42.732183032 -113.4286757 198.89304 -196.095241 281.55961
## 2005.6192   45.859841518 -110.4319939 202.15168 -193.167894 284.88758
## 2005.6219   34.320962963 -122.1017394 190.74367 -204.906916 273.54884
## 2005.6247   35.642283249 -120.9111768 192.19574 -203.785572 275.07014
## 2005.6274   43.788809439 -112.8952991 200.47292 -195.838856 283.41647
## 2005.6301   44.793740485 -112.0209076 201.60839 -195.033568 284.62105
## 2005.6329   49.288972020 -107.6561072 206.23405 -190.737814 289.31576
## 2005.6356   50.716090367 -106.3593116 207.79149 -189.510007 290.94219
## 2005.6384   47.224855576 -109.9807611 204.43047 -193.200388 287.65010
## 2005.6411   46.456338311 -110.8793853 203.79206 -194.167886 287.08056
## 2005.6438   40.166301714 -117.2994213 197.63202 -200.656740 280.98934
## 2005.6466   42.185966022 -115.4096492 199.78158 -198.835729 283.20766
## 2005.6493   42.209535683 -115.5158648 199.93494 -199.010648 283.42972
## 2005.6521   35.276997548 -122.5780815 193.13208 -206.141513 276.69551
## 2005.6548   38.997354432 -118.9872967 196.98201 -202.619319 280.61403
## 2005.6575   37.603466932 -120.5106501 195.71758 -204.211208 279.41814
## 2005.6603   30.207151695 -128.0363253 188.45063 -211.805362 272.21967
## 2005.6630   38.083964152 -120.2887672 196.45670 -204.126227 280.29416
## 2005.6658   47.178725172 -111.3231551 205.68061 -195.228982 289.58643
## 2005.6685   32.241154504 -126.3897695 190.87208 -210.363908 274.84622
## 2005.6712   33.585687527 -125.1741754 192.34555 -209.216570 276.38795
## 2005.6740   35.348971078 -123.5397261 194.23767 -207.650322 278.34826
## 2005.6767   28.895892713 -130.1215343 187.91332 -214.300275 272.09206
## 2005.6795   32.704518445 -126.4415343 191.85057 -210.688366 276.09740
## 2005.6822   38.271817692 -121.0027569 197.54639 -205.317624 281.86126
## 2005.6849   32.153293441 -127.2496994 191.55629 -211.632547 275.93913
## 2005.6877   32.204105051 -127.3272027 191.73541 -211.777976 276.18619
## 2005.6904   31.611480098 -128.0480394 191.27100 -212.566684 275.78964
## 2005.6932   24.213046428 -135.5745819 184.00067 -220.161043 268.58714
## 2005.6959   16.710352058 -143.2052826 176.62599 -227.859506 261.28021
## 2005.6986   28.063565923 -131.9799725 188.10710 -216.701904 272.82904
## 2005.7014   19.911454185 -140.2598860 180.08279 -225.049472 264.87238
## 2005.7041   21.487171019 -138.8118690 181.78621 -223.669055 266.64340
## 2005.7068   27.696886174 -132.7297520 188.12352 -217.654484 273.04826
## 2005.7096   31.816808737 -128.7373262 192.37094 -213.729551 277.36317
## 2005.7123   25.896362044 -134.7851686 186.57789 -219.844833 271.63756
## 2005.7151   18.590424341 -142.2184009 179.39925 -227.345451 264.52630
## 2005.7178    5.793477291 -155.1425420 166.72950 -240.336924 251.92388
## 2005.7205   12.524404879 -148.5387080 173.58752 -233.800370 258.84918
## 2005.7233   24.209971195 -136.9801350 185.40008 -222.309023 270.72897
## 2005.7260   33.806593656 -127.5104059 195.12359 -212.906467 280.51965
## 2005.7288   29.925832856 -131.5179603 191.36963 -216.981142 276.83281
## 2005.7315   23.399205628 -138.1712817 184.96969 -223.701531 270.49994
## 2005.7342   29.103283202 -132.5937990 190.80037 -218.191064 276.39763
## 2005.7370   17.404858666 -144.4187193 179.22844 -230.082947 264.89266
## 2005.7397   21.505138907 -140.4448361 183.45511 -226.175974 269.18625
## 2005.7425   35.756576275 -126.3196972 197.83285 -212.117694 283.63085
## 2005.7452   31.339167345 -130.8633062 193.54164 -216.728109 279.40644
## 2005.7479   28.088568082 -134.2400075 190.41714 -220.171565 276.34870
## 2005.7507   21.794251503 -140.6603282 184.24883 -226.658588 270.24709
## 2005.7534   20.460183792 -142.1203024 183.04067 -228.185213 269.10558
## 2005.7562   22.781944749 -139.9243504 185.48824 -226.055860 271.61975
## 2005.7589   24.238186315 -138.5938207 187.07019 -224.791878 273.26825
## 2005.7616   29.278887962 -133.6787339 192.23651 -219.943288 278.50106
## 2005.7644   23.788320851 -139.2948191 186.87146 -225.625819 273.20246
## 2005.7671   20.985190714 -142.2233708 184.19375 -228.620764 270.59115
## 2005.7699    8.816776303 -154.5171104 172.15066 -240.980847 258.61440
## 2005.7726   14.985407920 -148.4737080 178.44452 -235.003737 264.97455
## 2005.7753   25.567823645 -138.0164256 189.15207 -224.612696 275.74834
## 2005.7781   18.990750478 -144.7185364 182.70004 -231.380998 269.36250
## 2005.7808   21.234797135 -142.5994319 185.06903 -229.328034 271.79763
## 2005.7836   17.992357761 -145.9667183 181.95143 -232.761410 268.74613
## 2005.7863   23.189046005 -140.8947821 187.27287 -227.755514 274.13361
## 2005.7890    7.451932932 -156.7565524 171.66042 -243.683274 258.58714
## 2005.7918    2.653757020 -161.6792910 166.98681 -248.671952 253.97947
## 2005.7945    6.007289643 -158.4502267 170.46481 -245.508777 257.52336
## 2005.7973   16.180030526 -148.4018600 180.76192 -235.526250 267.88631
## 2005.8000   20.120778622 -144.5853922 184.82695 -231.775572 272.01713
## 2005.8027   20.734260823 -144.0960966 185.56462 -231.352017 272.82054
## 2005.8055   15.708147188 -149.2463033 180.66260 -236.567915 267.98421
## 2005.8082   15.368723119 -149.7097272 180.44717 -237.096980 267.83443
## 2005.8110   24.070679693 -141.1316773 189.27304 -228.584522 276.72588
## 2005.8137   21.659312035 -143.6668588 186.98548 -231.185247 274.50387
## 2005.8164   13.556988671 -151.8929034 179.00688 -239.476786 266.59076
## 2005.8192   20.110699154 -145.4628217 185.68422 -233.112149 273.33355
## 2005.8219    9.234050999 -156.4630063 174.93111 -244.177730 262.64583
## 2005.8247   13.105201055 -152.7153008 178.92570 -240.495372 266.70577
## 2005.8274   14.277208773 -151.6666457 180.22106 -239.512016 268.06643
## 2005.8301   17.026515479 -149.0406000 183.09363 -236.951221 271.00425
## 2005.8329   21.084056041 -145.1062291 187.27434 -233.082052 275.25016
## 2005.8356    5.349270103 -160.9640934 171.66263 -249.005070 259.70361
## 2005.8384   -2.109429416 -168.5457803 164.32692 -256.651862 252.43300
## 2005.8411   23.061021235 -143.4982262 189.62027 -231.669366 277.79141
## 2005.8438   28.751757721 -137.9302957 195.43381 -226.166445 283.66996
## 2005.8466   15.765399564 -151.0393694 182.57017 -239.340480 270.87128
## 2005.8493   10.873204646 -156.0541896 177.80060 -244.420214 266.16662
## 2005.8521   -8.822743830 -175.8726734 158.22719 -264.303564 246.65808
## 2005.8548    1.454257606 -165.7181174 168.62663 -254.213827 257.12234
## 2005.8575   10.643375125 -156.6513558 177.93811 -245.211837 266.49859
## 2005.8603   20.132413449 -147.2845839 187.54941 -235.909789 276.17462
## 2005.8630   30.246725480 -137.2924491 197.78590 -225.982331 286.47578
## 2005.8658   10.456461603 -157.2048012 178.11772 -245.959312 266.87224
## 2005.8685   10.121403216 -157.6618589 177.90467 -246.480953 266.72376
## 2005.8712    5.399322151 -162.5058507 173.30449 -251.389480 262.18812
## 2005.8740   13.395743794 -154.6312513 181.42274 -243.579370 270.37086
## 2005.8767   21.568312328 -146.5804167 189.71704 -235.592977 278.72960
## 2005.8795   16.667305525 -151.6030695 184.93768 -240.680025 274.01464
## 2005.8822    7.104670768 -161.2872623 175.49660 -250.428567 264.63791
## 2005.8849   11.588398610 -156.9250048 180.10180 -246.130612 269.30741
## 2005.8877   20.951828608 -147.6829577 189.58661 -236.952821 278.85648
## 2005.8904   19.720259780 -149.0358220 188.47634 -238.369895 277.81042
## 2005.8932   13.480305592 -155.3969847 182.35760 -244.795222 271.75583
## 2005.8959    6.037908877 -162.9605029 175.03632 -252.422858 264.49868
## 2005.8986    4.982663514 -164.1367830 174.10211 -253.663210 263.62854
## 2005.9014   -0.007340235 -169.2477350 169.23305 -258.838188 258.82351
## 2005.9041   17.584333165 -151.7769234 186.94559 -241.431357 276.60002
## 2005.9068   14.042840404 -155.4391918 183.52487 -245.157560 273.24324
## 2005.9096    0.193688408 -169.4090334 169.79641 -259.191291 259.57867
## 2005.9123   21.487598717 -148.2357269 191.21092 -238.081828 281.05703
## 2005.9151   35.502002978 -134.3418408 205.34585 -224.251741 295.25575
## 2005.9178   15.264535242 -154.6997413 185.22881 -244.673394 275.20246
## 2005.9205   13.287041452 -156.7975825 183.37167 -246.834944 273.40903
## 2005.9233   15.019356383 -155.1855299 185.22424 -245.286554 275.32527
## 2005.9260   10.462744577 -159.8623191 180.78781 -250.026961 270.95245
## 2005.9288    1.258169717 -169.1869867 171.70333 -259.415202 261.93154
## 2005.9315    4.015259069 -166.5499055 174.58042 -256.841649 264.87217
## 2005.9342  -17.337199570 -188.0222879 153.34789 -278.377516 243.70312
## 2005.9370    8.805058380 -161.9998695 179.60999 -252.418537 270.02865
## 2005.9397   22.703273752 -148.2214096 193.62796 -238.703471 284.11002
## 2005.9425   19.439372424 -151.6049827 190.48373 -242.150395 281.02914
## 2005.9452    8.365092064 -162.7988510 179.52904 -253.407569 270.13775
## 2005.9479    9.364797592 -161.9186500 180.64825 -252.590630 271.32023
## 2005.9507   17.513970255 -153.8888986 188.91684 -244.624096 279.65204
## 2005.9534   13.649091017 -157.8731159 185.17130 -248.671488 275.96967
## 2005.9562   16.359238260 -155.2822237 188.00070 -246.143725 278.86220
## 2005.9589    8.315842121 -163.4447922 180.07648 -254.369380 271.00106
## 2005.9616   11.913976055 -159.9657479 183.79370 -250.953378 274.78133
## 2005.9644   17.207938518 -154.7907926 189.20667 -245.841421 280.25730
## 2005.9671   15.218028182 -156.8996279 187.33568 -248.013211 278.44927
## 2005.9699    2.446018792 -169.7904801 174.68252 -260.966975 265.85901
## 2005.9726    6.112418176 -166.2428416 178.46768 -257.482205 269.70704
## 2005.9753    5.659208178 -166.8147306 178.13315 -258.116919 269.43534
## 2005.9781    3.537272747 -169.0552635 176.12981 -260.420233 267.49478
## 2005.9808   10.089589069 -162.6214633 182.80064 -254.049172 274.22835
## 2005.9836   12.231175991 -160.5983111 185.06066 -252.088715 276.55107
## 2005.9863   16.863428571 -156.0844122 189.81127 -247.637469 281.36433
## 2005.9890   15.401866603 -157.6642469 188.46798 -249.279913 280.08365
## 2005.9918   -1.723815351 -174.9081208 171.46049 -266.586354 263.13872
## 2005.9945   11.778424288 -161.5239925 185.08084 -253.264750 276.82160
## 2005.9973   13.395149732 -160.0252980 186.81560 -251.828538 278.61884
## 2006.0000   11.403125811 -162.1352725 184.94152 -254.000951 276.80720
## 2006.0027   32.225373152 -141.4308957 205.88164 -233.358971 297.80972
## 2006.0055   16.897052145 -156.8770072 190.67111 -248.867438 282.66154

##NOTE: AIC = 28718.12 , BIC = 28735.20 , RMSE = 14.87674

Also, the headline of the plot below contains ETS(A,N,N) showing that the automated model characterized error of the “tsWSR_PK” SERIES as Additative (A), trend as None (A) and seasonality as None (N) which is equivalent to simple exponential model(ses) with only alpha parameter i.e level.

#Model 2 : ARIMA Models(AutoRegressive Moving Average) It is used for forecasting for both seasonal and non- seasonal level.

#Changing to stationary set

train.sta_Var8 <- diff(train7)
test.sta_Var8 <- diff(test7)

#2.1 Auto Arima model without sesonality:

set.seed(301)
model_Var8 <- auto.arima(train.sta_Var8, seasonal = FALSE)
summary(model_Var8)
## Series: train.sta_Var8 
## ARIMA(2,0,3) with zero mean 
## 
## Coefficients:
##           ar1     ar2     ma1      ma2      ma3
##       -0.7108  0.2034  0.2693  -0.8819  -0.2156
## s.e.   0.0735  0.0493  0.0732   0.0211   0.0521
## 
## sigma^2 estimated as 236.6:  log likelihood=-9099.49
## AIC=18210.98   AICc=18211.02   BIC=18245.14
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set 0.004778148 15.36312 11.41507 Inf  Inf 0.6234888 0.001061865

NOTE: AIC = 18210.98 , BIC = 18245.14 , RMSE = 15.36312

tsdisplay(residuals(model_Var8), lag.max = 45, main = '(1,1,1) Model Residuals')

#2.2 Auto Arima model with sesonality:

set.seed(301)
model2_Var8 <- auto.arima(train.sta_Var8, seasonal= TRUE)
summary(model2_Var8)
## Series: train.sta_Var8 
## ARIMA(2,0,3) with zero mean 
## 
## Coefficients:
##           ar1     ar2     ma1      ma2      ma3
##       -0.7108  0.2034  0.2693  -0.8819  -0.2156
## s.e.   0.0735  0.0493  0.0732   0.0211   0.0521
## 
## sigma^2 estimated as 236.6:  log likelihood=-9099.49
## AIC=18210.98   AICc=18211.02   BIC=18245.14
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set 0.004778148 15.36312 11.41507 Inf  Inf 0.6234888 0.001061865

##NOTE: AIC = 18210.98 , BIC = 18245.14 , RMSE = 15.36312

tsdisplay(residuals(model2_Var8), lag.max = 20, main = 'Seasonal Model Residuals')

NOTE: Above Graphs shows serious lags at 13

#2.3 Manual Arima Passing parameter order = (2,0,13) which means p(member of lags of a variable to be used as a predictor) = 1, d = 0(members of differences needed for stationarity) and q = 18 (moving average order - number of lagged forecasts error in the prediction equation)

set.seed(301)
model1_Var8 <- arima(train.sta_Var8, order = c(2,0,13))
summary(model1_Var8)
## 
## Call:
## arima(x = train.sta_Var8, order = c(2, 0, 13))
## 
## Coefficients:
##           ar1      ar2      ma1     ma2      ma3      ma4     ma5      ma6
##       -0.0318  -0.3921  -0.4083  0.0161  -0.2453  -0.1824  0.0020  -0.0359
## s.e.   0.3417   0.2325   0.3409  0.2702   0.1662   0.0909  0.0306   0.0293
##         ma7      ma8     ma9     ma10    ma11     ma12    ma13  intercept
##       0.022  -0.0494  0.0428  -0.0130  0.0206  -0.0491  0.0575    -0.0002
## s.e.  0.027   0.0255  0.0269   0.0302  0.0262   0.0247  0.0238     0.0410
## 
## sigma^2 estimated as 234.8:  log likelihood = -9093.8,  aic = 18221.59
## 
## Training set error measures:
##                      ME     RMSE      MAE MPE MAPE     MASE          ACF1
## Training set 0.01190884 15.32315 11.36616 Inf  Inf 0.590761 -8.850224e-05

##NOTE: AIC = 18221.59 , RMSE = 15.32315

tsdisplay(residuals(model1_Var8), lag.max = 20, main = 'Seasonal Model Residuals')

Note: It seems good fit.

Fit the observed with the predicted values for both models in the same plot.

par(mfrow = c(2,2))
fit_auto_train_Var8 %>% forecast(h=341) %>% autoplot() 

model1_Var8 %>% forecast(h=341) %>% autoplot() 

model2_Var8 %>% forecast(h=341) %>% autoplot() 

model_Var8 %>% forecast(h=341) %>% autoplot() 

COMMENT:
1. The plots includes 80% to 95% confidence interval.
2. For ARIMA models it plots the forecast but because of the nature of autoregressive process of 2 lags, these forecast stabilizes quite smoothly.

Plot and comment on the residuals of the fitted data for both models in the same plot.

Residual Plot - to confirm no problem with this model

par(mfrow= c(2,2))
hist(fit_auto_train_Var8$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(fit_auto_train_Var8$residuals))

hist(model_Var8$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model_Var8$residuals))

hist(model1_Var8$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model1_Var8$residuals))

hist(model2_Var8$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model2_Var8$residuals))

OBSERVATION: The residuals for each model are normally distributed, which shows the good fit of the models. It is normally distrbuted.

Testing the autocorrelation and partial autocorrelation of the residuals (using the plots and Ljung-Box Q statistic) up to an appropriate lag *

  1. ACF & PACF Plots
par(mfrow = c(4,2))
acf(fit_auto_train_Var8$residuals, main = 'Correlogram')
pacf(fit_auto_train_Var8$residuals, main = 'Partial Corelogram')

acf(model_Var8$residuals, main = 'Correlogram')
pacf(model_Var8$residuals, main = 'Partial Corelogram')

acf(model1_Var8$residuals, main = 'Correlogram')
pacf(model1_Var8$residuals, main = 'Partial Corelogram')

acf(model2_Var8$residuals, main = 'Correlogram')
pacf(model2_Var8$residuals, main = 'Partial Corelogram')

NOTE:
1. Residuals are the difference between actaual and fitted values
2. These blue lines are signficance bounds
3. This graph shows us the autocorelations for insample forecast errors
4. Do not exceed these significance bounds for lags 1 to end

  1. Ljunx-Box Test to check the autocorelations of the residuals

HO: No serial correlation upto 20 lags (does not exhibit lack of fit of the model)
Ha: Serial corelation is present (the model exhibits lack of fit)

We reject the null hypothesis, if p value is less than 0.05. This test is generally done for ARIMA models

Box.test(model_Var8$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model_Var8$residuals
## X-squared = 27.528, df = 20, p-value = 0.1211
Box.test(model1_Var8$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model1_Var8$residuals
## X-squared = 9.6952, df = 20, p-value = 0.9734
Box.test(model2_Var8$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model2_Var8$residuals
## X-squared = 27.528, df = 20, p-value = 0.1211

OBSERVATIONS: For above performed Ljunx-Box Test on ARIMA models output, we can say that p-value is more than significance level 0.05, then there is not a statistical significance.
We conclude that there is a liitle evidence of non-zero auto-corelations in the insample forecast errors at lags 1 to 20.
So we fail to rejects the null hypothesis. Hence, we conclude that there is no serial auto corelation present between lags. Hence, the model seems to be good fit.

par(mfrow = c(2,2))
qqnorm(fit_auto_train_Var8$residuals)
qqnorm(model_Var8$residuals)
qqnorm(model1_Var8$residuals)
qqnorm(model2_Var8$residuals)

Validating The Model by testing the model performance with Holdout set

Here, choosing RMSE as evaluation parameter. The lower RMSE indicates a more accurate forecast.

Below, is the comparison of model. We are fitting the model to test set which is the holdout set. And checking the accuracy using RMSE(Root mean square error) evaluation parameter.

accuracy(forecast(fit_auto_train_Var8), test7) ["Test set", "RMSE"]
## [1] 22.98142
#o/p - 22.98142
accuracy(forecast(model1_Var8), test.sta_Var8) ["Test set", "RMSE"]
## [1] 16.92509
#O/P - 16.92509
accuracy(forecast(model2_Var8), test.sta_Var8) ["Test set", "RMSE"]
## [1] 16.9194
#o/p - 16.9194
accuracy(forecast(model_Var8), test.sta_Var8) ["Test set", "RMSE"]
## [1] 16.9194
#O/P - 16.9194

NOTE:- Here, the RMSE value is less of ARIMA models compared to exponential model.
The minimum RMSE is of model2_Var8

COMMENT ON BEST MODEL AND ITS PARAMETERS

We observed the best model to be ARIMA model i.e.“model2_Var8”.

summary(model2_Var8)
## Series: train.sta_Var8 
## ARIMA(2,0,3) with zero mean 
## 
## Coefficients:
##           ar1     ar2     ma1      ma2      ma3
##       -0.7108  0.2034  0.2693  -0.8819  -0.2156
## s.e.   0.0735  0.0493  0.0732   0.0211   0.0521
## 
## sigma^2 estimated as 236.6:  log likelihood=-9099.49
## AIC=18210.98   AICc=18211.02   BIC=18245.14
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set 0.004778148 15.36312 11.41507 Inf  Inf 0.6234888 0.001061865

Best model is ARIMA(2,0,3) ,which means

p = 2 d = 0 q = 3

For the best model, 2 member of lag is used as a predictor. As we transformed the series to stationary so members of differences needed for stationarity is 0. 3 are the number of lagged forecasts errors used in the prediction equation.
AIC = 18210.98, RMSE = 15.36312 which is minimum as compared to other models.

9. Forecasting on T-Totals

#Checking the attributes of time series

attributes(tsTT)
## $tsp
## [1] 1998.00 2004.94  365.00
## 
## $class
## [1] "ts"

#Plotting the series

plot(tsTT, main = " T-Totals ", col = "red")

From the plot, we observes that there seems to have no trend but seems to have seasonlity.

Checking the statistics and visualising the missing data in the series:

plotNA.distribution(tsTT)

As data is missing at random. Hence, using na_locf method to impute the missing data.

#Impute the missing values with “na_locf” function of “imputeTS package” and visualise the imputed values in the time series

tsTT.imp <- na_locf(tsTT)
plotNA.imputations(tsTT, tsTT.imp)

#Decomposition Of Time Series

decomp8 <- decompose(tsTT.imp)
plot(decomp8, col = "red")

OBSERVATIONS:
1)There seems to have trend.
2)We observes the upward trend from year 2000 to 2003 and then further downward trend till 2004.
3)We observe some seasonality in the series. Seasonality describes cyclical effects due to the time of the year.

Let’s explore this seasonality pattern by plotting the decomp$seasonal components such that y-axis represents Seasonality Index and x-axis has month.

plot(decomp8$seasonal[1:365], type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")

OBSERVATIONS:
1) There is increase in TT for mid month of the year
2) There is a huge drop of TT for starting and end months of the year.

Stationarity check of time series

We could check the stationarity of time series by ADF and KPSS Tests:

  1. Augmented Dickey- Fuller(ADF) t-statistic test for unit root
  2. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) for level or trend stationarity
adf.test(tsTT.imp)
## Warning in adf.test(tsTT.imp): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  tsTT.imp
## Dickey-Fuller = -6.9802, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary

#P-value is smaller so it is stationary.

kpss.test(tsTT.imp, null="Trend")
## 
##  KPSS Test for Trend Stationarity
## 
## data:  tsTT.imp
## KPSS Trend = 0.12999, Truncation lag parameter = 8, p-value = 0.07966

#P-value is greater than 0.05, which shows trend statationary.

Second, we can check each for characteristics of stationarity by looking at the autocorrelation functions (ACF) of each signal. For a stationary signal, because we expect no dependence with time, we would expect the ACF to go to 0 for each time lag (τ). Lets visualize the signals and ACFs.

acf(tsTT.imp,lag.max = length(tsTT.imp),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Here, we observes most of the data lies outside the blue-dotted lines means the significance level.
Now, we will take the difference to make our data stationary.

It seems difference of 1 is sufficient to make the series stationary.

tsTT.sta <- diff(tsTT.imp)
acf(tsTT.sta, lag.max = length(tsTT.sta),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Hence, after differencing our series is now transformed to stationary time series.

#CREATING MODELS AND FORECASTING: USING EXPONENTIAL SMOOTHING AND ARIMA TECHNIQUES FOR TIME SERIES FORECASTING

1.Train - Test Split of the data

train8 <- window(tsTT.imp, end = c(2004, 3))
test8 <- window(tsTT.imp, start = c(2004, 4))

#Model 1 : Auto Exponential Smoothening Model

set.seed(301)
fit_auto_train_Var9 <- forecast(train8)
summary(fit_auto_train_Var9)
## 
## Forecast method: STL +  ETS(A,N,N)
## 
## Model Information:
## ETS(A,N,N) 
## 
## Call:
##  ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend) 
## 
##   Smoothing parameters:
##     alpha = 0.1081 
## 
##   Initial states:
##     l = 41.5637 
## 
##   sigma:  8.9365
## 
##      AIC     AICc      BIC 
## 26480.80 26480.81 26497.88 
## 
## Error measures:
##                       ME     RMSE      MAE  MPE MAPE      MASE      ACF1
## Training set 0.003227413 8.932469 6.564425 -Inf  Inf 0.6349621 0.3549989
## 
## Forecasts:
##           Point Forecast         Lo 80    Hi 80         Lo 95     Hi 95
## 2004.0082       38.05206  26.599417775 49.50470  20.536754953  55.56737
## 2004.0110       44.13985  32.620459208 55.65923  26.522464120  61.75723
## 2004.0137       43.71898  32.133232347 55.30473  26.000108537  61.43785
## 2004.0164       45.50479  33.853056831 57.15651  27.685004362  63.32457
## 2004.0192       31.34877  19.631431669 43.06611  13.428647226  49.26889
## 2004.0219       35.67864  23.896057690 47.46122  17.658734672  53.69855
## 2004.0247       44.40569  32.558224073 56.25316  26.286552683  62.52483
## 2004.0274       33.64943  21.737433140 45.56144  15.431600475  51.86727
## 2004.0301       43.38445  31.408262045 55.36063  25.068452175  61.70044
## 2004.0329       37.24568  25.205647338 49.28570  18.832041391  55.65931
## 2004.0356       29.63456  17.531030310 41.73810  11.123806546  48.14532
## 2004.0384       40.10417  27.937462816 52.27088  21.496796706  58.71154
## 2004.0411       45.43561  33.206056083 57.66517  26.732120377  64.13910
## 2004.0438       45.14160  32.849515531 57.43368  26.342480330  63.94071
## 2004.0466       30.50848  18.154191007 42.86278  11.614223828  49.40274
## 2004.0493       41.67156  29.255369553 54.08775  22.682635397  60.66048
## 2004.0521       36.67938  24.201599640 49.15716  17.596261051  55.76250
## 2004.0548       27.72467  15.185602976 40.26374   8.547820103  46.90153
## 2004.0575       29.70543  17.105365870 42.30549  10.435296526  48.97556
## 2004.0603       36.68991  24.029155249 49.35067  17.326954966  56.05287
## 2004.0630       31.46150  18.740331913 44.18266  12.006153996  50.91684
## 2004.0658       40.85418  28.072891495 53.63546  21.306887076  60.40147
## 2004.0685       39.19012  26.348994611 52.03125  19.551312698  58.82893
## 2004.0712       36.89381  23.993124903 49.79450  17.163912432  56.62372
## 2004.0740       33.53295  20.572969702 46.49293  13.712371582  53.35352
## 2004.0767       37.58838  24.569383057 50.60738  17.677542218  57.49922
## 2004.0795       44.89515  31.817396862 57.97290  24.894454299  64.89584
## 2004.0822       47.73714  34.600895642 60.87337  27.646990457  67.82728
## 2004.0849       40.45803  27.263562801 53.65250  20.278832248  60.63723
## 2004.0877       40.58408  27.331632291 53.83652  20.316211812  60.85194
## 2004.0904       43.20534  29.895169910 56.51550  22.849193180  63.56148
## 2004.0932       47.38589  34.018252483 60.75353  26.941851444  67.82993
## 2004.0959       39.10380  25.678933871 52.52867  18.572238771  59.63536
## 2004.0986       35.42049  21.938637309 48.90234  14.801776736  56.03920
## 2004.1014       35.44482  21.906223542 48.98341  14.739324462  56.15031
## 2004.1041       38.81612  25.221019801 52.41122  18.024207589  59.60804
## 2004.1068       41.03925  27.387870372 54.69062  20.161268847  61.91722
## 2004.1096       43.53861  29.831194818 57.24603  22.574926274  64.50230
## 2004.1123       42.82059  29.057355513 56.58382  21.771540749  63.86963
## 2004.1151       40.62689  26.808070445 54.44571  19.492828798  61.76095
## 2004.1178       45.29577  31.421580820 59.16995  24.077030191  66.51450
## 2004.1205       44.76894  30.839608400 58.69827  23.465865285  66.07202
## 2004.1233       25.56919  11.584929658 39.55345   4.182109175  46.95627
## 2004.1260       29.75068  15.711706585 43.78966   8.279922499  51.22144
## 2004.1288       43.71397  29.620494686 57.80745  22.159859439  65.26808
## 2004.1315       45.04934  30.901569266 59.19710  23.412194000  66.68648
## 2004.1342       42.04024  27.838394020 56.24209  20.320388602  63.76010
## 2004.1370       33.65704  19.401308994 47.91277  11.854782042  55.45929
## 2004.1397       36.61034  22.300937982 50.91975  14.725996885  58.49469
## 2004.1425       40.82219  26.459308578 55.18507  18.856059522  62.78832
## 2004.1452       47.26170  32.845539452 61.67785  25.214087441  69.30930
## 2004.1479       44.71297  30.243733078 59.18221  22.584181956  66.84176
## 2004.1507       49.21724  34.695114434 63.73936  27.007566907  71.42691
## 2004.1534       43.63344  29.058621168 58.20826  21.343178824  65.92370
## 2004.1562       40.03627  25.408943803 54.66359  17.665707132  62.40682
## 2004.1589       44.96472  30.285077589 59.64436  22.514146002  67.41529
## 2004.1616       37.45906  22.727289795 52.19083  14.928761644  59.98936
## 2004.1644       38.03190  23.248183129 52.81562  15.422155726  60.64165
## 2004.1671       47.28856  32.453076082 62.12404  24.599645717  69.97747
## 2004.1699       45.43330  30.546231724 60.32037  22.665493683  68.20111
## 2004.1726       47.98295  33.044472988 62.92142  25.136521567  70.82938
## 2004.1753       48.06944  33.079736897 63.05915  25.144665426  70.99422
## 2004.1781       44.71185  29.671082826 59.75261  21.708983679  67.71471
## 2004.1808       46.31142  31.219774009 61.40307  23.230738621  69.39210
## 2004.1836       33.95589  18.813532667 49.09825  10.797651554  57.11413
## 2004.1863       35.10157  19.908662164 50.29447  11.866024936  58.33711
## 2004.1890       35.40598  20.162705601 50.64926  12.093400974  58.71857
## 2004.1918       33.34715  18.053665295 48.64064   9.957781108  56.73653
## 2004.1945       43.19716  27.853621061 58.54069  19.731244294  66.66307
## 2004.1973       47.63298  32.239560023 63.02640  24.090776805  71.17518
## 2004.2000       44.74159  29.298450862 60.18473  21.123346488  68.35983
## 2004.2027       48.03858  32.545875147 63.53128  24.344534091  71.73262
## 2004.2055       52.22363  36.681522793 67.76574  28.454028722  75.99323
## 2004.2082       49.88108  34.289723037 65.47243  26.036158820  73.72600
## 2004.2110       46.61843  30.977984539 62.25888  22.698432264  70.53843
## 2004.2137       43.18428  27.494890145 58.87366  19.189431130  67.17912
## 2004.2164       36.38216  20.643991218 52.12034  12.312706020  60.45162
## 2004.2192       41.22820  25.441388862 57.01501  17.084357294  65.37204
## 2004.2219       46.48482  30.649522366 62.32011  22.266823504  70.70281
## 2004.2247       46.27708  30.393447445 62.16071  21.985159641  70.56900
## 2004.2274       43.85501  27.923180462 59.78683  19.489381354  68.22063
## 2004.2301       40.99337  25.013501296 56.97325  16.554267822  65.43248
## 2004.2329       43.45054  27.422765133 59.47832  18.938173537  67.96291
## 2004.2356       43.16291  27.087378908 59.23845  18.577504752  67.74832
## 2004.2384       40.15470  24.031547594 56.27785  15.496465771  64.81294
## 2004.2411       47.49680  31.326171648 63.66743  22.765956387  72.22765
## 2004.2438       50.90476  34.686793247 67.12273  26.101518126  75.70801
## 2004.2466       53.31754  37.052366886 69.58271  28.442104841  78.19297
## 2004.2493       51.21627  34.904033500 67.52851  26.268856833  76.16368
## 2004.2521       45.32464  28.965474270 61.68381  20.305454659  70.34383
## 2004.2548       48.02115  31.615186172 64.42711  22.930394681  73.11190
## 2004.2575       45.30211  28.849488798 61.75474  20.139995884  70.46423
## 2004.2603       37.18714  20.687983307 53.68629  11.953858828  62.42042
## 2004.2630       41.42110  24.875547379 57.96665  16.116860605  66.72534
## 2004.2658       40.70115  24.109322741 57.29297  15.326142361  66.07615
## 2004.2685       45.73603  29.098065876 62.37399  20.290460006  71.18160
## 2004.2712       40.05556  23.371586613 56.73954  14.539622802  65.57150
## 2004.2740       41.39062  24.660752854 58.12048  15.804498095  66.97673
## 2004.2767       42.69612  25.920495519 59.47174  17.040016256  68.35222
## 2004.2795       37.77019  20.948929844 54.59145  12.044291978  63.49609
## 2004.2822       45.87669  29.009911098 62.74346  20.081179995  71.67219
## 2004.2849       47.48571  30.573547618 64.39788  21.620788117  73.35064
## 2004.2877       49.01629  32.058851095 65.97372  23.082127513  74.95044
## 2004.2904       44.38267  27.380084469 61.38525  18.379460610  70.38587
## 2004.2932       46.88631  29.838699255 63.93392  20.814238415  72.95838
## 2004.2959       44.40388  27.311359304 61.49640  18.263124281  70.54464
## 2004.2986       49.12578  31.988462571 66.26309  22.916515667  75.33504
## 2004.3014       52.08333  34.901342722 69.26533  25.805745754  78.36092
## 2004.3041       49.44831  32.221753311 66.67486  23.102567613  75.79404
## 2004.3068       48.44035  31.169355041 65.71135  22.026641473  74.85406
## 2004.3096       45.52237  28.207037607 62.83769  19.040856561  72.00387
## 2004.3123       49.53341  32.173860103 66.89295  22.984271508  76.08254
## 2004.3151       46.13875  28.735096005 63.54240  19.522159332  72.75534
## 2004.3178       47.07126  29.623615919 64.51891  20.387390190  73.75513
## 2004.3205       47.05234  29.560810141 64.54387  20.301353931  73.80332
## 2004.3233       47.02329  29.487984021 64.55859  20.205355466  73.84122
## 2004.3260       50.02484  32.445873941 67.60381  23.140130743  76.90955
## 2004.3288       51.98681  34.364285948 69.60933  25.035485378  78.93813
## 2004.3315       44.28756  26.621589108 61.95353  17.269788016  71.30534
## 2004.3342       37.22445  19.515135599 54.93377  10.140390417  64.30851
## 2004.3370       46.14720  28.394643350 63.89975  18.997010094  73.29738
## 2004.3397       49.93302  32.137336599 67.72870  22.716870877  77.14917
## 2004.3425       46.54110  28.702390668 64.37981  19.259147687  73.82305
## 2004.3452       41.28752  23.405885509 59.16915  13.939920076  68.63512
## 2004.3479       40.25724  22.332779925 58.18169  12.844146453  67.67032
## 2004.3507       44.70684  26.739668001 62.67402  17.228420514  72.18526
## 2004.3534       48.49030  30.480506707 66.50009  20.946698845  76.03390
## 2004.3562       44.13151  26.079199763 62.18382  16.522884787  71.74013
## 2004.3589       40.69984  22.605109922 58.79456  13.026340715  68.37333
## 2004.3616       52.27126  34.134212016 70.40830  24.533041093  80.00947
## 2004.3644       48.30986  30.130600146 66.48913  20.507079652  76.11265
## 2004.3671       47.92910  29.707719067 66.15049  20.061900788  75.79631
## 2004.3699       48.03246  29.769049880 66.29587  20.100985241  75.96393
## 2004.3726       46.66880  28.363463646 64.97414  18.673203719  74.66440
## 2004.3753       49.59329  31.246121281 67.94046  21.533716788  77.65286
## 2004.3781       50.46473  32.075825671 68.85364  22.341326988  78.58814
## 2004.3808       47.55460  29.124056689 65.98515  19.367513848  75.74170
## 2004.3836       45.22935  26.757253685 63.70145  16.978716382  73.47998
## 2004.3863       50.87390  32.360347107 69.38745  22.559864702  79.18793
## 2004.3890       50.85345  32.298538161 69.40837  22.476159683  79.23075
## 2004.3918       46.97410  28.377914444 65.57029  18.533688596  75.41451
## 2004.3945       47.51414  28.876774538 66.15150  19.010749699  76.01753
## 2004.3973       49.13797  30.459518558 67.81642  20.571742786  77.70420
## 2004.4000       50.62915  31.909703603 69.34861  22.000224641  79.25809
## 2004.4027       50.39761  31.637247345 69.15797  21.706112621  79.08910
## 2004.4055       48.81289  30.011714536 67.61407  20.058971172  77.56682
## 2004.4082       48.64098  29.799064778 67.48289  19.824759586  77.45719
## 2004.4110       48.05093  29.168379804 66.93349  19.172559295  76.92931
## 2004.4137       44.79901  25.875902110 63.72212  15.858612495  73.73941
## 2004.4164       44.52847  25.564891744 63.49205  15.526178937  73.53076
## 2004.4192       48.34381  29.339845905 67.34777  19.279755528  77.40786
## 2004.4219       48.11349  29.069232301 67.15775  18.987809683  77.23918
## 2004.4247       52.62444  33.539962668 71.70891  23.437252854  81.81162
## 2004.4274       51.26988  32.145279165 70.39448  22.021326915  80.51843
## 2004.4301       49.88853  30.723885070 69.05317  20.578734862  79.19833
## 2004.4329       47.79433  28.589726158 66.99894  18.423422192  77.16524
## 2004.4356       47.74031  28.495825722 66.98479  18.308411922  77.17221
## 2004.4384       47.84330  28.559025647 67.12758  18.350545667  77.33606
## 2004.4411       48.48126  29.157270872 67.80525  18.927768093  78.03476
## 2004.4438       46.74444  27.380815639 66.10806  17.130333178  76.35854
## 2004.4466       48.21585  28.812676391 67.61902  18.541257100  77.89044
## 2004.4493       50.18783  30.745191913 69.63048  20.452878380  79.92279
## 2004.4521       50.50706  31.025026590 69.98909  20.711861148  80.30226
## 2004.4548       49.80402  30.282674128 69.32536  19.948698852  79.65934
## 2004.4575       49.42998  29.869405932 68.99056  19.514662642  79.34530
## 2004.4603       48.88649  29.286762356 68.48622  18.911292622  78.86169
## 2004.4630       48.21846  28.579652011 67.85726  18.183497155  78.25341
## 2004.4658       48.20442  28.526616053 67.88222  18.109817151  78.29902
## 2004.4685       50.44060  30.723878480 70.15732  20.286476363  80.59472
## 2004.4712       51.12620  31.370629630 70.88176  20.912664888  81.33973
## 2004.4740       48.68358  28.889251569 68.47792  18.410764554  78.95640
## 2004.4767       49.86503  30.032006076 69.69806  19.533036902  80.19702
## 2004.4795       48.78316  28.911517949 68.65480  18.392106496  79.17421
## 2004.4822       47.71027  27.800091067 67.62046  17.260276985  78.16027
## 2004.4849       48.95910  29.010448013 68.90775  18.450270719  79.46792
## 2004.4877       50.28805  30.301004347 70.27509  19.720503033  80.85559
## 2004.4904       47.14507  27.119704561 67.17043  16.518918192  77.77122
## 2004.4932       45.37714  25.313536307 65.44075  14.692503625  76.06179
## 2004.4959       48.13629  28.034503916 68.23807  17.393263442  78.87931
## 2004.4986       49.50418  29.364295822 69.64406  18.702885858  80.30547
## 2004.5014       48.08339  27.905480557 68.26130  17.223939189  78.94285
## 2004.5041       47.07993  26.864056971 67.29580  16.162422069  77.99743
## 2004.5068       49.22253  28.968772260 69.47628  18.247081481  80.19798
## 2004.5096       48.86524  28.573668620 69.15681  17.831959411  79.89852
## 2004.5123       46.58730  26.257981449 66.91662  15.496291046  77.67831
## 2004.5151       49.91359  29.546597487 70.28058  18.764962921  81.06222
## 2004.5178       50.67173  30.267134239 71.07633  19.465592335  81.87787
## 2004.5205       51.13840  30.696260657 71.58053  19.874848037  82.40194
## 2004.5233       50.76686  30.287258715 71.24646  19.446011799  82.08771
## 2004.5260       49.18333  28.666330191 69.70033  17.805285202  80.56138
## 2004.5288       48.84676  28.292423184 69.40109  17.411616144  80.28190
## 2004.5315       49.37314  28.781537937 69.96473  17.881004673  80.86527
## 2004.5342       50.24134  29.612545583 70.87013  18.692321729  81.79036
## 2004.5370       49.48282  28.816892451 70.14874  17.877013448  81.08862
## 2004.5397       49.21796  28.514978298 69.92095  17.555479396  80.88045
## 2004.5425       49.54475  28.804771406 70.28474  17.825687666  81.26382
## 2004.5452       47.80338  27.026463204 68.58029  16.027829499  79.57892
## 2004.5479       49.97938  29.165602550 70.79316  18.147453569  81.81131
## 2004.5507       48.53662  27.686045930 69.38720  16.648416178  80.42483
## 2004.5534       48.28804  27.400727795 69.17536  16.343651594  80.23243
## 2004.5562       47.18766  26.263671904 68.11164  15.187183394  79.18813
## 2004.5589       49.47571  28.515116351 70.43630  17.419249495  81.53217
## 2004.5616       48.23248  27.235346919 69.22962  16.120135501  80.34483
## 2004.5644       46.39473  25.361115438 67.42834  14.226593065  78.56286
## 2004.5671       48.77722  27.707185728 69.84724  16.553385836  81.00104
## 2004.5699       49.73392  28.627534742 70.84030  17.454490589  82.01334
## 2004.5726       49.91538  28.772710990 71.05806  17.580455667  82.25031
## 2004.5753       50.09602  28.917116651 71.27492  17.705683076  82.48635
## 2004.5781       51.36684  30.151774458 72.58191  18.921195382  83.81249
## 2004.5808       48.15113  26.899953988 69.40230  15.650261993  80.65199
## 2004.5836       49.29017  28.002952421 70.57739  16.734179925  81.84616
## 2004.5863       49.63898  28.315783255 70.96219  17.027962510  82.25001
## 2004.5890       45.87819  24.519066317 67.23731  13.212229415  78.54415
## 2004.5918       49.47633  28.081346587 70.87132  16.755525454  82.19714
## 2004.5945       47.87939  26.448600746 69.31018  15.103827152  80.65495
## 2004.5973       49.86734  28.400807435 71.33387  17.037112987  82.69756
## 2004.6000       51.28499  29.782774918 72.78720  18.400191069  84.16978
## 2004.6027       49.79736  28.259526007 71.33520  16.858084051  82.73664
## 2004.6055       46.90040  25.327003321 68.47381  13.906734399  79.89408
## 2004.6082       48.17280  26.563894992 69.78171  15.124830090  81.22078
## 2004.6110       46.08592  24.441560466 67.73027  12.983730417  79.18810
## 2004.6137       48.61488  26.935135098 70.29463  15.458570585  81.77119
## 2004.6164       49.04702  27.331939699 70.76210  15.836671254  82.25737
## 2004.6192       49.24238  27.492029952 70.99274  15.978087960  82.50668
## 2004.6219       41.76949  19.983918580 63.55506   8.451333277  75.08765
## 2004.6247       45.75414  23.933401596 67.57487  12.382203071  79.12607
## 2004.6274       47.11822  25.262378502 68.97405  13.692596700  80.54384
## 2004.6301       48.29373  26.402839530 70.18461  14.814504251  81.77295
## 2004.6329       50.64259  28.716709835 72.56847  17.109850737  84.17533
## 2004.6356       50.96636  29.005545044 72.92718  17.380191643  84.55253
## 2004.6384       49.48269  27.486990779 71.47838  15.843172449  83.12220
## 2004.6411       49.68256  27.652033121 71.71308  15.989779099  83.37533
## 2004.6438       47.63864  25.573346084 69.70393  13.892685467  81.38459
## 2004.6466       49.61110  27.511088344 71.71111  15.812050091  83.41015
## 2004.6493       48.50768  26.373006402 70.64235  14.655619337  82.35974
## 2004.6521       48.05292  25.883644445 70.22220  14.147937256  81.95791
## 2004.6548       49.13792  26.934089401 71.34175  15.180090643  83.09575
## 2004.6575       46.39750  24.159172477 68.63584  12.386910571  80.40810
## 2004.6603       42.72889  20.456112498 65.00167   8.665615733  76.79217
## 2004.6630       45.64738  23.340212991 67.95456  11.531509524  79.76326
## 2004.6658       48.49685  26.155338955 70.83836  14.328456815  82.66525
## 2004.6685       42.28204  19.906236115 64.65784   8.061203201  76.50287
## 2004.6712       44.41372  22.003685415 66.82376  10.140529498  78.68691
## 2004.6740       45.87790  23.433684335 68.32212  11.552433058  80.20337
## 2004.6767       38.69111  16.212756971 61.16945   4.313437852  73.06877
## 2004.6795       41.22192  18.709490515 63.73435   6.792130947  75.65171
## 2004.6822       43.88480  21.338339919 66.43125   9.402967169  78.36662
## 2004.6849       41.61278  19.032345773 64.19321   7.078986988  76.14657
## 2004.6877       38.35397  15.739610328 60.96833   3.768292529  72.93964
## 2004.6904       40.14876  17.500528136 62.79699   5.511278226  74.78624
## 2004.6932       38.95567  16.273610723 61.63772   4.266455480  73.64488
## 2004.6959       34.24959  11.533756997 56.96542  -0.491276915  68.99045
## 2004.6986       43.15792  20.408364038 65.90747   8.365477998  77.95036
## 2004.7014       36.08008  13.296852929 58.86331   1.236141185  70.92402
## 2004.7041       36.37878  13.561928004 59.19563   1.483416864  71.27414
## 2004.7068       38.61642  15.765990719 61.46684   3.669706374  73.56313
## 2004.7096       43.87718  20.993226186 66.76113   8.879194713  78.87516
## 2004.7123       41.47336  18.555933029 64.39079   6.424180389  76.52254
## 2004.7151       35.86079  12.909938576 58.81165   0.760490618  70.96109
## 2004.7178       34.91067  11.926435874 57.89490  -0.240681669  70.06202
## 2004.7205       38.20614  15.188572925 61.22370   3.003811423  73.40846
## 2004.7233       43.17130  20.120452169 66.22214   7.918072218  78.42452
## 2004.7260       46.49246  23.408384848 69.57654  11.188411851  81.79651
## 2004.7288       44.01490  20.897630321 67.13216   8.660089570  79.36970
## 2004.7315       35.38183  12.231428800 58.53224  -0.023654521  70.78732
## 2004.7342       41.16998  17.986484639 64.35347   5.713883822  76.62608
## 2004.7370       30.42040   7.203861315 53.63694  -5.086232028  65.92703
## 2004.7397       32.39939   9.149852258 55.64892  -3.157708751  67.95649
## 2004.7425       41.97294  18.690448772 65.25542   6.365444855  77.58043
## 2004.7452       43.17363  19.858240177 66.48902   7.515818002  78.83144
## 2004.7479       42.78089  19.432643709 66.12914   7.072827823  78.48896
## 2004.7507       40.05617  16.675115109 63.43723   4.297929956  75.81442
## 2004.7534       39.00978  15.595954439 62.42360   3.201424359  74.81813
## 2004.7562       44.20856  20.762015448 67.65510   8.350164680  80.06695
## 2004.7589       37.85204  14.372825528 61.33126   1.943678209  73.76041
## 2004.7616       39.70472  16.192870008 63.21656   3.746450175  75.66298
## 2004.7644       38.59131  15.046877901 62.13574   2.583209490  74.59941
## 2004.7671       35.28048  11.703515779 58.85745  -0.777377371  71.33835
## 2004.7699       28.16824   4.558773492 51.77770  -7.939320660  64.27579
## 2004.7726       40.15278  16.510867564 63.79469   3.995596053  76.30996
## 2004.7753       41.32734  17.653023898 65.00165   5.120598571  77.53408
## 2004.7781       34.82174  11.115065532 58.52842  -1.434490165  71.07797
## 2004.7808       39.17960  15.440613500 62.91860   2.873950786  75.48526
## 2004.7836       37.57790  13.806640410 61.34917   1.222893935  73.93291
## 2004.7863       36.22293  12.419435052 60.02642  -0.181372023  72.62722
## 2004.7890       28.20041   4.364739111 52.03609  -8.253105496  64.65394
## 2004.7918       25.77186   1.904047693 49.63968 -10.730811472  62.27454
## 2004.7945       30.20640   6.306489505 54.10632  -6.345361336  66.75817
## 2004.7973       35.98256  12.050591900 59.91453  -0.618227827  72.58335
## 2004.8000       39.63831  15.674329079 63.60229   2.988563163  76.28806
## 2004.8027       40.50417  16.508217635 64.50012   3.805528138  77.20281
## 2004.8055       37.55319  13.525312145 61.58107   0.805721584  74.30066
## 2004.8082       35.18123  11.121467988 59.24099  -1.615001210  71.97746
## 2004.8110       32.81643   8.724825729 56.90804  -4.028499768  69.66136
## 2004.8137       42.60917  18.485764083 66.73258   5.715604538  79.50273
## 2004.8164       37.40565  13.250483764 61.56081   0.463512333  74.34778
## 2004.8192       41.48322  17.296341012 65.67010   4.492579769  78.47386
## 2004.8219       29.39635   5.177792249 53.61490  -7.642736818  66.43543
## 2004.8247       36.66305  12.412857399 60.91324  -0.424417590  73.75051
## 2004.8274       37.14791  12.866125195 61.42969   0.012126100  74.28369
## 2004.8301       43.05347  18.740135754 67.36680   5.869434285  80.23750
## 2004.8329       40.39577  16.050928690 64.74062   3.163546493  77.62800
## 2004.8356       26.05467   1.678351812 50.43098 -11.225689550  63.33502
## 2004.8384       24.81776   0.410019116 49.22551 -12.510659933  62.14619
## 2004.8411       38.73874  14.299605112 63.17787   1.362309774  76.11517
## 2004.8438       44.05393  19.583451719 68.52441   6.629561406  81.47830
## 2004.8466       32.61054   8.108753291 57.11233  -4.861710765  70.08280
## 2004.8493       30.75888   6.225825108 55.29194  -6.761191540  68.27896
## 2004.8521       23.08254  -1.481747674 47.64683 -14.485295843  60.65037
## 2004.8548       31.22730   6.631822924 55.82278  -6.388235778  68.84283
## 2004.8575       32.26842   7.641791024 56.89504  -5.394757298  69.93159
## 2004.8603       40.64795  15.990215182 65.30569   2.937198069  78.35870
## 2004.8630       45.58187  20.893062320 70.27068   7.823597168  83.34014
## 2004.8658       34.01025   9.290409061 58.73009  -3.795483455  71.81598
## 2004.8685       34.26583   9.514993880 59.01666  -3.587305404  72.11896
## 2004.8712       32.74793   7.966143182 57.52972  -5.152542351  70.64840
## 2004.8740       34.63272   9.820021620 59.44543  -3.315029719  72.58048
## 2004.8767       42.42436  17.580776286 67.26794   4.429379505  80.41933
## 2004.8795       37.69305  12.818627636 62.56747  -0.349094296  75.73519
## 2004.8822       30.19451   5.289293120 55.09973  -7.894733748  68.28376
## 2004.8849       36.92234  11.986358091 61.85832  -1.213953573  75.05863
## 2004.8877       41.84988  16.883173658 66.81659   3.666597262  80.03316
## 2004.8904       39.62682  14.629423573 64.62421   1.396602438  77.85703
## 2004.8932       35.64712  10.619072328 60.67516  -2.629973630  73.92421
## 2004.8959       35.61822  10.559564233 60.67688  -2.705686702  73.94213
## 2004.8986       33.41806   8.328832132 58.50729  -4.952604009  71.78873
## 2004.9014       29.13459   4.014820345 54.25436  -9.282781301  67.55196
## 2004.9041       36.38628  11.236011959 61.53655  -2.077735565  74.85029
## 2004.9068       33.40616   8.225425138 58.58689  -5.104448707  71.91676
## 2004.9096       32.19031   6.979157111 57.40147  -6.366823569  70.74745
## 2004.9123       40.26730  15.025756823 65.50885   1.663688723  78.87092
## 2004.9151       47.97051  22.698605801 73.24241   9.320469627  86.62054
## 2004.9178       41.01660  15.714380797 66.31882   2.320195825  79.71300
## 2004.9205       34.66353   9.331034337 59.99603  -4.079180227  73.40625
## 2004.9233       37.60420  12.241455105 62.96694  -1.184769913  76.39317
## 2004.9260       29.73436   4.341412908 55.12732  -9.100803495  68.56953
## 2004.9288       28.05550   2.632376332 53.47862 -10.825812454  66.93681
## 2004.9315       33.36584   7.912584225 58.81911  -5.561558011  72.29325
## 2004.9342       19.47529  -6.008076007 44.95865 -19.498152826  58.44872
## 2004.9370       32.52023   7.006804036 58.03366  -6.499188567  71.53965
## 2004.9397       39.93585  14.392388262 65.47930   0.870498610  79.00119
## 2004.9425       36.81478  11.241326000 62.38823  -2.296442034  75.92600
## 2004.9452       34.62007   9.016657063 60.22348  -4.536970751  73.77711
## 2004.9479       33.51171   7.878368552 59.14504  -5.691100506  72.71451
## 2004.9507       36.17356  10.510336780 61.83679  -3.074955049  75.42208
## 2004.9534       31.48523   5.792143858 57.17831  -7.808952336  70.77940
## 2004.9562       34.96719   9.244283522 60.69009  -4.372598693  74.30697
## 2004.9589       29.21710   3.464416182 54.96979 -10.168233775  68.60244
## 2004.9616       32.79734   7.014903355 58.57978  -6.633496128  72.22818
## 2004.9644       36.00588  10.193726334 61.81804  -3.470404522  75.48217
## 2004.9671       35.89997  10.058124564 61.74181  -3.621719573  75.42165
## 2004.9699       30.40431   4.532817822 56.27580  -9.162721569  69.97134
## 2004.9726       29.42862   3.527511581 55.32972 -10.183705097  69.04094
## 2004.9753       32.24617   6.315479820 58.17685  -7.411396240  71.90373
## 2004.9781       33.39260   7.432367026 59.35283  -6.310150573  73.09535
## 2004.9808       32.23033   6.240578128 58.22007  -7.517563227  71.97821
## 2004.9836       36.43058  10.411354183 62.44981  -3.362393206  76.22356
## 2004.9863       36.94264  10.893960111 62.99131  -2.895375650  76.78065
## 2004.9890       33.79272   7.714631305 59.87081  -6.090275225  73.67572
## 2004.9918       24.27567  -1.831798247 50.38314 -15.652258003  64.20360
## 2004.9945       34.58791   8.451089014 60.72472  -5.384906486  74.56072
## 2004.9973       35.88607   9.719939499 62.05220  -4.131574320  75.90372
## 2005.0000       37.52110  11.325684208 63.71651  -2.541330563  77.58353
## 2005.0027       44.58918  18.364512872 70.81384   4.482014457  84.69634
## 2005.0055       34.13236   7.878484228 60.38625  -6.019480581  74.28421
## 2005.0082       38.05206  11.768995759 64.33513  -2.144418251  78.24854
## 2005.0110       44.13985  17.827629562 70.45206   3.898783487  84.38091
## 2005.0137       43.71898  17.377642835 70.06032   3.433381773  84.00458
## 2005.0164       45.50479  19.134361677 71.87521   5.174702650  85.83487
## 2005.0192       31.34877   4.949291377 57.74825  -9.025748650  71.72329
## 2005.0219       35.67864   9.250138864 62.10714  -4.740265253  76.09755
## 2005.0247       44.40569  17.948199246 70.86319   3.942447895  84.86894
## 2005.0274       33.64943   7.162980616 60.13589  -6.858101172  74.15697
## 2005.0301       43.38445  16.869065735 69.89983   2.832670254  83.93623
## 2005.0329       37.24568  10.701396610 63.78995  -3.350295874  77.84165
## 2005.0356       29.63456   3.061419845 56.20771 -11.005553009  70.27468
## 2005.0384       40.10417  13.502192464 66.70615  -0.580044179  80.78839
## 2005.0411       45.43561  18.804830730 72.06639   4.707346826  86.16388
## 2005.0438       45.14160  18.482044968 71.80115   4.369330274  85.91386
## 2005.0466       30.50848   3.820189801 57.19678 -10.307739262  71.32471
## 2005.0493       41.67156  14.954556931 68.38856   0.811429865  82.53169
## 2005.0521       36.67938   9.933699364 63.42506  -4.224609390  77.58337
## 2005.0548       27.72467   0.950343237 54.49900 -13.223130945  68.67248
## 2005.0575       29.70543   2.902479174 56.50837 -11.286144225  70.69700
## 2005.0603       36.68991   9.858378311 63.52145  -4.345378147  77.72520
## 2005.0630       31.46150   4.601405559 58.32159  -9.617467854  72.54046
## 2005.0658       40.85418  13.965560557 67.74280  -0.268413755  81.97677
## 2005.0685       39.19012  12.273007837 66.10723  -1.976051371  80.35629
## 2005.0712       36.89381   9.948234861 63.83939  -4.315893290  78.10352
## 2005.0740       33.53295   6.558932691 60.50696  -7.720248499  74.78614
## 2005.0767       37.58838  10.585959023 64.59080  -3.708259356  78.88502
## 2005.0795       44.89515  17.864349310 71.92594   3.555109545  86.23518
## 2005.0822       47.73714  20.677991558 74.79628   6.353746159  89.12052
## 2005.0849       40.45803  13.370572572 67.54549  -0.968662757  81.88473
## 2005.0877       40.58408  13.468329628 67.69982  -0.885879978  82.05403
## 2005.0904       43.20534  16.061331776 70.34934   1.692163498  84.71851
## 2005.0932       47.38589  20.213659019 74.55813   5.829547626  88.94224
## 2005.0959       39.10380  11.903368329 66.30423  -2.495670672  80.70327
## 2005.0986       35.42049   8.191885982 62.64909  -6.222065168  77.06304
## 2005.1014       35.44482   8.188075700 62.70156  -6.240772187  77.13041
## 2005.1041       38.81612  11.531267627 66.10098  -2.912461633  80.54471
## 2005.1068       41.03925  13.726308900 68.35218  -0.732286416  82.81078
## 2005.1096       43.53861  16.197621876 70.87960   1.724175773  85.35305
## 2005.1123       42.82059  15.451571659 70.18960   0.963289992  84.67788
## 2005.1151       40.62689  13.229878916 68.02390  -1.273223141  82.52700
## 2005.1178       45.29577  17.870787474 72.72075   3.352880157  87.23865
## 2005.1205       44.76894  17.316021664 72.22186   2.783324170  86.75456
## 2005.1233       25.56919  -1.911639525 53.05002 -16.459112159  67.59749
## 2005.1260       29.75068   2.241968363 57.25939 -12.320264419  71.82163
## 2005.1288       43.71397  16.177403253 71.25054   1.600425267  85.82752
## 2005.1315       45.04934  17.484942816 72.61373   2.893234528  87.20544
## 2005.1342       42.04024  14.448053072 69.63244  -0.158370665  84.23886
## 2005.1370       33.65704   6.037076343 61.27700  -8.584048031  75.89812
## 2005.1397       36.61034   8.962638657 64.25805  -5.673171589  78.89386
## 2005.1425       40.82219  13.146769798 68.49761  -1.503711598  83.14809
## 2005.1452       47.26170  19.558590585 74.96480   4.893452716  89.62994
## 2005.1479       44.71297  16.982205600 72.44373   2.302425891  87.12351
## 2005.1507       49.21724  21.458841889 76.97563   6.764434930  91.67004
## 2005.1534       43.63344  15.847439131 71.41944   1.138419467  86.12846
## 2005.1562       40.03627  12.222689841 67.84984  -2.500928025  82.57346
## 2005.1589       44.96472  17.123591223 72.80584   2.385389615  87.54404
## 2005.1616       37.45906   9.590412467 65.32771  -5.162358467  80.08048
## 2005.1644       38.03190  10.135758166 65.92804  -4.631567720  80.69537
## 2005.1671       47.28856  19.364948661 75.21217   4.583082155  89.99404
## 2005.1699       45.43330  17.482248840 73.38435   2.685856003  88.18074
## 2005.1726       47.98295  20.004483421 75.96141   5.193578500  90.77232
## 2005.1753       48.06944  20.063591181 76.07530   5.238188382  90.90070
## 2005.1781       44.71185  16.678633219 72.74506   1.838746705  87.58495
## 2005.1808       46.31142  18.250874459 74.37197   3.396518353  89.22632
## 2005.1836       33.95589   5.868038788 62.04375  -9.000772830  76.91256
## 2005.1863       35.10157   6.986431204 63.21670  -7.896821885  78.09995
## 2005.1890       35.40598   7.263596414 63.54837  -7.634084147  78.44605
## 2005.1918       33.34715   5.177538314 61.51677  -9.734555760  76.42886
## 2005.1945       43.19716  15.000338274 71.39397   0.073844605  86.32047
## 2005.1973       47.63298  19.408984942 75.85697   4.468105555  90.79785
## 2005.2000       44.74159  16.490448501 72.99273   1.535197235  87.94798
## 2005.2027       48.03858  19.760311996 76.31684   4.790702648  91.28645
## 2005.2055       52.22363  23.918266794 80.52899   8.934313123  95.51294
## 2005.2082       49.88108  21.548643559 78.21351   6.550359284  93.21179
## 2005.2110       46.61843  18.258952356 74.97791   3.246351156  89.99051
## 2005.2137       43.18428  14.797777413 71.57077  -0.229127071  86.59768
## 2005.2164       36.38216   7.968671452 64.79566  -7.072522714  79.83685
## 2005.2192       41.22820  12.787736915 69.66866  -2.267733371  84.72413
## 2005.2219       46.48482  18.017414408 74.95222   2.947681527  90.02195
## 2005.2247       46.27708  17.782760941 74.77140   2.698778950  89.85538
## 2005.2274       43.85501  15.333794151 72.37622   0.235576499  87.47444
## 2005.2301       40.99337  12.445295175 69.54145  -2.667144730  84.65389
## 2005.2329       43.45054  14.875620431 72.02546  -0.251028354  87.15211
## 2005.2356       43.16291  14.561178072 71.76465  -0.579666258  86.90549
## 2005.2384       40.15470  11.526174270 68.78323  -3.628852309  83.93825
## 2005.2411       47.49680  18.841510660 76.15210   3.672315090  91.32129
## 2005.2438       50.90476  22.222730580 79.58680   7.039379244  94.77015
## 2005.2466       53.31754  24.608789670 82.02629   9.411295752  97.22378
## 2005.2493       51.21627  22.480829992 79.95171   7.269206641  95.16333
## 2005.2521       45.32464  16.562533834 74.08675   1.336794162  89.31249
## 2005.2548       48.02115  19.232399267 76.80990   3.992556349  92.04974
## 2005.2575       45.30211  16.486746961 74.11748   1.232813837  89.37141
## 2005.2603       37.18714   8.345179134 66.02910  -6.922831193  81.29711
## 2005.2630       41.42110  12.552574511 70.28963  -2.729500051  85.57170
## 2005.2658       40.70115  11.806075852 69.59622  -3.490050014  84.89234
## 2005.2685       45.73603  16.814440654 74.65762   1.504276380  89.96778
## 2005.2712       40.05556  11.107479745 69.00365  -4.216710076  84.32784
## 2005.2740       41.39062  12.416062016 70.36517  -2.922140528  85.70337
## 2005.2767       42.69612  13.695119357 71.69712  -1.657083118  87.04932
## 2005.2795       37.77019   8.742767964 66.79761  -6.623421689  82.16380
## 2005.2822       45.87669  16.822864049 74.93051   1.442699940  90.31067
## 2005.2849       47.48571  18.405516882 76.56591   3.011391002  91.96003
## 2005.2877       49.01629  19.909739071 78.12283   4.501664072  93.53091
## 2005.2904       44.38267  15.249794463 73.51554  -0.172217039  88.93755
## 2005.2932       46.88631  17.727135465 76.04549   2.291200043  91.48142
## 2005.2959       44.40388  15.218426810 73.58934  -0.231419983  89.03918
## 2005.2986       49.12578  19.914067319 78.33749   4.450321669  93.80123
## 2005.3014       52.08333  22.845391519 81.32128   7.367759494  96.79891
## 2005.3041       49.44831  20.184153805 78.71246   4.692647852  94.20396
## 2005.3068       48.44035  19.150015716 77.73069   3.644648249  93.23606
## 2005.3096       45.52237  16.205867769 74.83886   0.686651170  90.35808
## 2005.3123       49.53341  20.190769870 78.87604   4.657716485  94.40910
## 2005.3151       46.13875  16.769996294 75.50750   1.223118439  91.05438
## 2005.3178       47.07126  17.676418438 76.46610   2.115728395  92.02679
## 2005.3205       47.05234  17.631427378 76.47325   2.056937395  92.04774
## 2005.3233       47.02329  17.576329233 76.47024   1.988051528  92.05852
## 2005.3260       50.02484  20.551861143 79.49782   4.949807900  95.09987
## 2005.3288       51.98681  22.487829905 81.48579   6.872013277  97.10161
## 2005.3315       44.28756  14.762605326 73.81252  -0.866962569  89.44209
## 2005.3342       37.22445   7.673540313 66.77536  -7.969766759  82.41867
## 2005.3370       46.14720  16.570353515 75.72404   0.913319322  91.38107
## 2005.3397       49.93302  20.330269882 79.53577   4.659520592  95.20652
## 2005.3425       46.54110  16.912465439 76.16974   1.228013045  91.85419
## 2005.3452       41.28752  11.633020830 70.94202  -4.065122706  86.64016
## 2005.3479       40.25724  10.576895545 69.93757  -5.134927203  85.64940
## 2005.3507       44.70684  15.000684343 74.41300  -0.724805718  90.13849
## 2005.3534       48.49030  18.758344863 78.22225   3.019199357  93.96140
## 2005.3562       44.13151  14.373781483 73.88924  -1.379007630  89.64202
## 2005.3589       40.69984  10.916357608 70.48331  -4.850063305  86.24973
## 2005.3616       52.27126  22.462048714 82.08046   6.682007776  97.86050
## 2005.3644       48.30986  18.474949535 78.14478   2.681300318  93.93843
## 2005.3671       47.92910  18.068505455 77.78970   2.261259674  93.59695
## 2005.3699       48.03246  18.146198195 77.91872   2.325367536  93.73955
## 2005.3726       46.66880  16.756899428 76.58070   0.922495544  92.41511
## 2005.3753       49.59329  19.655770674 79.53081   3.807805191  95.37878
## 2005.3781       50.46473  20.501615417 80.42785   4.640099931  96.28936
## 2005.3808       47.55460  17.565914121 77.54330   1.690860196  93.41835
## 2005.3836       45.22935  15.215106719 75.24359  -0.673474108  91.13217
## 2005.3863       50.87390  20.834124235 80.91367   4.932028012  96.81577
## 2005.3890       50.85345  20.788168445 80.91874   4.872568303  96.83434
## 2005.3918       46.97410  16.883327509 77.06487   0.954234897  92.99396
## 2005.3945       47.51414  17.397900565 77.63038   1.455326901  93.57295
## 2005.3973       49.13797  18.996288277 79.27965   3.040244951  95.23570
## 2005.4000       50.62915  20.462048290 80.79626   4.492546663  96.76576
## 2005.4027       50.39761  20.205098809 80.59012   4.222150214  96.57306
## 2005.4055       48.81289  18.595005121 79.03078   2.598620861  95.02717
## 2005.4082       48.64098  18.397727351 78.88422   2.387918702  94.89403
## 2005.4110       48.05093  17.782347749 78.31952   1.759125959  94.34274
## 2005.4137       44.79901  14.505109327 75.09292  -1.531514387  91.12954
## 2005.4164       44.52847  14.209272638 74.84767  -1.840741809  90.89769
## 2005.4192       48.34381  17.999335383 78.68828   1.935941367  94.75168
## 2005.4219       48.11349  17.743765764 78.48322   1.667003313  94.55998
## 2005.4247       52.62444  22.229476009 83.01940   6.139356231  99.10952
## 2005.4274       51.26988  20.849708762 81.69005   4.746242735  97.79352
## 2005.4301       49.88853  19.443167778 80.33389   3.326366555  96.45069
## 2005.4329       47.79433  17.323799306 78.26486   1.193673911  94.39499
## 2005.4356       47.74031  17.244627109 78.23599   1.101188540  94.37943
## 2005.4384       47.84330  17.322493535 78.36411   1.165752761  94.52085
## 2005.4411       48.48126  17.935343980 79.02718   1.765311945  95.19721
## 2005.4438       46.74444  16.173433141 77.31544  -0.009879239  93.49876
## 2005.4466       48.21585  17.619777909 78.81192   1.423196073  95.00850
## 2005.4493       50.18783  19.566717511 80.80895   3.356877081  97.01879
## 2005.4521       50.50706  19.860916773 81.15320   3.637828586  97.37629
## 2005.4548       49.80402  19.132869834 80.47517   2.896544698  96.71149
## 2005.4575       49.42998  18.733848528 80.12611   2.484297225  96.37567
## 2005.4603       48.88649  18.165393632 79.60759   1.902626921  95.87036
## 2005.4630       48.21846  17.472414180 78.96450   1.196442790  95.24047
## 2005.4658       48.20442  17.433451744 78.97538   1.144286379  95.26455
## 2005.4685       50.44060  19.644730731 81.23647   3.342382070  97.53882
## 2005.4712       51.12620  20.305441887 81.94695   3.989920582  98.26247
## 2005.4740       48.68358  17.837967680 79.52920   1.509284358  95.85789
## 2005.4767       49.86503  18.994570289 80.73549   2.652735550  97.07733
## 2005.4795       48.78316  17.887874905 79.67844   1.532899325  96.03342
## 2005.4822       47.71027  16.790185799 78.63036   0.422079928  94.99847
## 2005.4849       48.95910  18.014225937 79.90397   1.633000300  96.28520
## 2005.4877       50.28805  19.318411265 81.25768   2.924076360  97.65202
## 2005.4904       47.14507  16.150686650 78.13945  -0.256747048  94.54688
## 2005.4932       45.37714  14.358040120 76.39625  -2.062481922  92.81677
## 2005.4959       48.13629  17.092476377 79.18009   0.658876415  95.61369
## 2005.4986       49.50418  18.435684221 80.57267   1.989016737  97.01934
## 2005.5014       48.08339  16.990232548 79.17655   0.530507918  95.63628
## 2005.5041       47.07993  15.962120566 78.19773  -0.510650861  94.67050
## 2005.5068       49.22253  18.080095826 80.36496   1.594287928  96.85077
## 2005.5096       48.86524  17.698200880 80.03228   1.199366811  96.53111
## 2005.5123       46.58730  15.395671472 77.77893  -1.116178492  94.29078
## 2005.5151       49.91359  18.697394687 81.12979   2.172539080  97.65464
## 2005.5178       50.67173  19.430988372 81.91248   2.893137350  98.45033
## 2005.5205       51.13840  19.873121818 82.40367   3.322285586  98.95451
## 2005.5233       50.76686  19.477077334 82.05665   2.913266071  98.62046
## 2005.5260       49.18333  17.869057031 80.49761   1.292280891  97.07438
## 2005.5288       48.84676  17.508009334 80.18550   0.918278451  96.77524
## 2005.5315       49.37314  18.009934812 80.73634   1.407259294  97.33901
## 2005.5342       50.24134  18.853704921 81.62897   2.238094853  98.24458
## 2005.5370       49.48282  18.070766309 80.89487   1.442231752  97.52340
## 2005.5397       49.21796  17.781519049 80.65441   1.140070040  97.29586
## 2005.5425       49.54475  18.083931734 81.00558   1.429578289  97.65993
## 2005.5452       47.80338  16.318196105 79.28856  -0.349051786  95.95581
## 2005.5479       49.97938  18.469861326 81.48890   1.789728957  98.16903
## 2005.5507       48.53662  17.002784187 80.07047   0.309777285  96.76347
## 2005.5534       48.28804  16.729899442 79.84618   0.024027928  96.55206
## 2005.5562       47.18766  15.605231145 78.77008  -1.113495081  95.48881
## 2005.5589       49.47571  17.869017687 81.08240   1.137446625  97.81397
## 2005.5616       48.23248  16.601545143 79.86342  -0.142860901  96.60782
## 2005.5644       46.39473  14.739565634 78.04989  -2.017665563  94.80712
## 2005.5671       48.77722  17.097843267 80.45659   0.327796727  97.22663
## 2005.5699       49.73392  18.030355277 81.43748   1.247503179  98.22033
## 2005.5726       49.91538  18.187650460 81.64312   1.392002568  98.43877
## 2005.5753       50.09602  18.344131271 81.84791   1.535697325  98.65634
## 2005.5781       51.36684  19.590820721 83.14287   2.769610440  99.96408
## 2005.5808       48.15113  16.350988660 79.95127  -0.482988259  96.78524
## 2005.5836       49.29017  17.465932541 81.11441   0.619198659  97.96114
## 2005.5863       49.63898  17.790666129 81.48730   0.931184937  98.34678
## 2005.5890       45.87819  14.005809520 77.75057  -2.866409351  94.62279
## 2005.5918       49.47633  17.579907956 81.37276   0.694961014  98.25770
## 2005.5945       47.87939  15.958938382 79.79984  -0.938727043  96.69750
## 2005.5973       49.86734  17.922879696 81.81179   1.012505354  98.72217
## 2005.6000       51.28499  19.316540421 83.25343   2.393466706 100.17651
## 2005.6027       49.79736  17.804943623 81.78978   0.869180057  98.72555
## 2005.6055       46.90040  14.884032174 78.91678  -2.064411742  95.86522
## 2005.6082       48.17280  16.132494455 80.21311  -0.828620330  97.17423
## 2005.6110       46.08592  14.021690160 78.15014  -2.952086035  95.12392
## 2005.6137       48.61488  16.526754891 80.70301  -0.459673277  97.68944
## 2005.6164       49.04702  16.935009701 81.15903  -0.064061023  98.15810
## 2005.6192       49.24238  17.106510517 81.37826   0.094806631  98.38996
## 2005.6219       41.76949   9.609770299 73.92921  -7.414557373  90.95354
## 2005.6247       45.75414  13.570585298 77.93768  -3.466356807  94.97463
## 2005.6274       47.11822  14.910855251 79.32558  -2.138691953  96.37513
## 2005.6301       48.29373  16.062570622 80.52488  -0.999572369  97.58702
## 2005.6329       50.64259  18.387656799 82.89752   1.312927312  99.97225
## 2005.6356       50.96636  18.687669636 83.24505   1.600362926 100.33236
## 2005.6384       49.48269  17.180254984 81.78512   0.080380300  98.88499
## 2005.6411       49.68256  17.356399148 82.00871   0.243965722  99.12115
## 2005.6438       47.63864  15.288776366 79.98850  -1.836206594  97.11349
## 2005.6466       49.61110  17.237545533 81.98465   0.100022230  99.12217
## 2005.6493       48.50768  16.110453373 80.90490  -1.039601104  98.05496
## 2005.6521       48.05292  15.632044286 80.47380  -1.530532214  97.63638
## 2005.6548       49.13792  16.693405419 81.58244  -0.481683976  98.75753
## 2005.6575       46.39750  13.929368192 78.86564  -3.258224987  96.05323
## 2005.6603       42.72889  10.237151641 75.22063  -6.962936233  92.42072
## 2005.6630       45.64738  13.132059504 78.16271  -4.080513995  95.37528
## 2005.6658       48.49685  15.957956988 81.03575  -1.267093086  98.26080
## 2005.6685       42.28204   9.719590026 74.84448  -7.517927593  92.08200
## 2005.6712       44.41372  11.827739765 76.99970  -5.422236387  94.24968
## 2005.6740       45.87790  13.268403889 78.48740  -3.994021805  95.74983
## 2005.6767       38.69111   6.058106694 71.32411 -11.216759569  88.59897
## 2005.6795       41.22192   8.565435576 73.87840  -8.721862304  91.16570
## 2005.6822       43.88480  11.204845680 76.56475  -6.094874883  93.86447
## 2005.6849       41.61278   8.909377797 74.31618  -8.402756536  91.62831
## 2005.6877       38.35397   5.627134370 71.08080 -11.697404838  88.40534
## 2005.6904       40.14876   7.398510146 72.89901  -9.938425060  90.23595
## 2005.6932       38.95567   6.182016840 71.72932 -11.167305507  89.07864
## 2005.6959       34.24959   1.452553555 67.04662 -15.909147097  84.40832
## 2005.6986       43.15792  10.337517555 75.97832  -7.036552581  93.35239
## 2005.7014       36.08008   3.236330113 68.92383 -14.150100708  86.31026
## 2005.7041       36.37878   3.511695746 69.24586 -13.887086978  86.64464
## 2005.7068       38.61642   5.726016097 71.50682 -11.685109767  88.91794
## 2005.7096       43.87718  10.963476459 76.79088  -6.459983801  94.21434
## 2005.7123       41.47336   8.536375637 74.41034  -8.899410294  91.84613
## 2005.7151       35.86079   2.900541139 68.82104 -14.547561755  86.26915
## 2005.7178       34.91067   1.927166190 67.89417 -15.533244979  85.35458
## 2005.7205       38.20614   5.199398969 71.21287 -12.273311804  88.68558
## 2005.7233       43.17130  10.141342092 76.20125  -7.343659634  93.68625
## 2005.7260       46.49246  13.439306974 79.54562  -4.057977071  97.04290
## 2005.7288       44.01490  10.938553147 77.09124  -6.571004601  94.60079
## 2005.7315       35.38183   2.282320995 68.48134 -15.239501859  86.00317
## 2005.7342       41.16998   8.047315040 74.29264  -9.486764340  91.82672
## 2005.7370       30.42040  -2.725401070 63.56620 -20.271728415  81.11253
## 2005.7397       32.39939  -0.769533740 65.56831 -18.328100506  83.12688
## 2005.7425       41.97294   8.780908502 75.16496  -8.789889160  92.73576
## 2005.7452       43.17363   9.958515140 76.38875  -7.624504910  93.97177
## 2005.7479       42.78089   9.542703571 76.01908  -8.052530375  93.61431
## 2005.7507       40.05617   6.794929702 73.31742 -10.812509669  90.92486
## 2005.7534       39.00978   5.725493754 72.29406 -11.894142587  89.91370
## 2005.7562       44.20856  10.901249636 77.51587  -6.730575238  95.14769
## 2005.7589       37.85204   4.521724897 71.18236 -13.122280090  88.82637
## 2005.7616       39.70472   6.351405025 73.05803 -11.304771673  90.71421
## 2005.7644       38.59131   5.215019188 71.96760 -12.453320835  89.63594
## 2005.7671       35.28048   1.881234113 68.67973 -15.799260867  86.36023
## 2005.7699       28.16824  -5.253960196 61.59043 -22.946601784  79.28307
## 2005.7726       40.15278   6.707652937 73.59790 -10.997126924  91.30268
## 2005.7753       41.32734   7.859299566 74.79538  -9.857610252  92.51229
## 2005.7781       34.82174   1.330802879 68.31268 -16.398228597  86.04171
## 2005.7808       39.17960   5.665784061 72.69342 -12.075360792  90.43457
## 2005.7836       37.57790   4.041215865 71.11459 -13.712034099  88.86784
## 2005.7863       36.22293   2.663387229 69.78247 -15.101959597  87.54781
## 2005.7890       28.20041  -5.381960016 61.78279 -23.159395473  79.56023
## 2005.7918       25.77186  -7.833330619 59.37706 -25.622846493  77.16658
## 2005.7945       30.20640  -3.421595730 63.83441 -21.223183823  81.63599
## 2005.7973       35.98256   2.331772146 69.63335 -15.481879984  87.44700
## 2005.8000       39.63831   5.964747351 73.31188 -11.860960651  91.13758
## 2005.8027       40.50417   6.807846620 74.20049 -11.029909107  92.03825
## 2005.8055       37.55319   3.834124667 71.27226 -14.015670653  89.12205
## 2005.8082       35.18123   1.439437011 68.92303 -16.422389786  86.78485
## 2005.8110       32.81643  -0.948075646 66.58094 -18.821925822  84.45479
## 2005.8137       42.60917   8.821965547 76.39637  -9.063899925  94.28224
## 2005.8164       37.40565   3.595761440 71.21553 -14.302111263  89.11341
## 2005.8192       41.48322   7.650668407 75.31578 -10.259203477  93.22565
## 2005.8219       29.39635  -4.458856998 63.25155 -22.380720028  81.17342
## 2005.8247       36.66305   2.785205282 70.54089 -15.148640877  88.47474
## 2005.8274       37.14791   3.247444112 71.04837 -14.698377174  88.99419
## 2005.8301       43.05347   9.130399739 76.97654  -8.827388689  94.93433
## 2005.8329       40.39577   6.450111907 74.34143 -11.519635693  92.31118
## 2005.8356       26.05467  -7.913571448 60.02290 -25.895270266  78.00460
## 2005.8384       24.81776  -9.173036202 58.80856 -27.166678301  76.80220
## 2005.8411       38.73874   4.725392284 72.75208 -13.280185173  90.75766
## 2005.8438       44.05393  10.018056053 78.08981  -7.999448857  96.10731
## 2005.8466       32.61054  -1.447850418 66.66894 -19.477274888  84.69836
## 2005.8493       30.75888  -3.322011721 64.83978 -21.363347878  82.88111
## 2005.8521       23.08254 -11.020842579 57.18592 -29.074082563  75.23916
## 2005.8548       31.22730  -2.898554892 65.35315 -20.963690859  83.41829
## 2005.8575       32.26842  -1.879894413 66.41673 -19.956918534  84.49375
## 2005.8603       40.64795   6.477197531 74.81871 -11.611706932  92.90761
## 2005.8630       45.58187  11.388687982 79.77505  -6.712089025  97.87583
## 2005.8658       34.01025  -0.205346315 68.22584 -18.317988084  86.33848
## 2005.8685       34.26583   0.027833230 68.50382 -18.096665533  86.62832
## 2005.8712       32.74793  -1.512446860 67.00831 -19.648794866  85.14465
## 2005.8740       34.63272   0.349978185 68.91547 -17.798211328  87.06366
## 2005.8767       42.42436   8.119255572 76.72946 -10.040767726  94.88948
## 2005.8795       37.69305   3.365605872 72.02049 -14.806243505  90.19234
## 2005.8822       30.19451  -4.155253351 64.54428 -22.338921115  82.72795
## 2005.8849       36.92234   2.550263368 71.29442 -15.645215107  89.48990
## 2005.8877       41.84988   7.455507252 76.24425 -10.751774273  94.45154
## 2005.8904       39.62682   5.210162165 74.04347 -13.008914763  92.26255
## 2005.8932       35.64712   1.208192708 70.08604 -17.022671992  88.31690
## 2005.8959       35.61822   1.157043302 70.07940 -17.085601552  88.32204
## 2005.8986       33.41806  -1.065353099 67.90148 -19.319770506  86.15589
## 2005.9014       29.13459  -5.371052067 63.64023 -23.637234439  81.90641
## 2005.9041       36.38628   1.858429592 70.91413 -16.419510173  89.19207
## 2005.9068       33.40616  -1.143889848 67.95620 -19.433579448  86.24589
## 2005.9096       32.19031  -2.381913056 66.76254 -20.683344946  85.06397
## 2005.9123       40.26730   5.672909023 74.86170 -12.640257628  93.17487
## 2005.9151       47.97051  13.353958019 82.58706  -4.970935880 100.91195
## 2005.9178       41.01660   6.377910788 75.65529 -11.958702858  93.99190
## 2005.9205       34.66353   0.002719961 69.32435 -18.345605946  87.67267
## 2005.9233       37.60420   2.921274323 72.28712 -15.438756373  90.64715
## 2005.9260       29.73436  -4.970656216 64.43938 -23.342384244  82.81111
## 2005.9288       28.05550  -6.671602968 62.78260 -25.055020885  81.16602
## 2005.9315       33.36584  -1.383326984 68.11502 -19.778427362  86.51012
## 2005.9342       19.47529 -15.295940760 54.24651 -33.702716183  72.65329
## 2005.9370       32.52023  -2.273035793 67.31350 -20.691478862  85.73194
## 2005.9397       39.93585   5.120551921 74.75114 -13.309551406  93.18124
## 2005.9425       36.81478   1.977471812 71.65209 -16.464284402  90.09384
## 2005.9452       34.62007  -0.239236213 69.47938 -18.692637955  87.93278
## 2005.9479       33.51171  -1.369584954 68.39300 -19.834624878  86.85804
## 2005.9507       36.17356   1.270301999 71.07683 -17.206368778  89.55350
## 2005.9534       31.48523  -3.439993149 66.41045 -21.928287462  84.89874
## 2005.9562       34.96719   0.020023433 69.91435 -18.479887112  88.41426
## 2005.9589       29.21710  -5.751987749 64.18620 -24.263507238  82.69772
## 2005.9616       32.79734  -2.193665086 67.78835 -20.716786242  86.31147
## 2005.9644       36.00588   0.992972809 71.01880 -17.541742752  89.55351
## 2005.9671       35.89997   0.865165474 70.93477 -17.681137244  89.48107
## 2005.9699       30.40431  -4.652367222 65.46098 -23.210249862  84.01887
## 2005.9726       29.42862  -5.649919715 64.50715 -24.219375057  83.07661
## 2005.9753       32.24617  -2.854217935 67.34655 -21.435238771  85.92757
## 2005.9781       33.39260  -1.729617306 68.51482 -20.322196440  87.10740
## 2005.9808       32.23033  -2.913712808 67.37436 -21.517843060  85.97849
## 2005.9836       36.43058   1.264736704 71.59643 -17.350937499  90.21210
## 2005.9863       36.94264   1.754996240 72.13027 -16.872214759  90.75749
## 2005.9890       33.79272  -1.416698720 69.00214 -20.055439375  87.64088
## 2005.9918       24.27567 -10.955514100 59.50686 -29.605777283  78.15712
## 2005.9945       34.58791  -0.665032255 69.84085 -19.326810852  88.50262
## 2005.9973       35.88607   0.611393312 71.16075 -18.061893597  89.83404
## 2006.0000       37.52110   2.224693687 72.81750 -16.460094446  91.50229
## 2006.0027       44.58918   9.271058687 79.90729  -9.425223594  98.60358
## 2006.0055       34.13236  -1.207452867 69.47218 -19.915222235  88.17995

##NOTE: AIC = 26480.80 , BIC = 26497.88 , RMSE = 8.932469

Also, the headline of the plot below contains ETS(A,N,N) showing that the automated model characterized error of the “tsWSR_PK” SERIES as Additative (A), trend as None (A) and seasonality as None (N) which is equivalent to simple exponential model(ses) with only alpha parameter i.e level.

#Model 2 : ARIMA Model(AutoRegressive Moving Average)

It is used for forecasting for both seasonal and non- seasonal level.

#Changing to stationary set

train.sta_Var9 <- diff(train8)
test.sta_Var9 <- diff(test8)

#2.1 Auto Arima model without sesonality:

set.seed(301)
model_Var9 <- auto.arima(train.sta_Var9, seasonal = FALSE)
summary(model_Var9)
## Series: train.sta_Var9 
## ARIMA(2,0,1) with zero mean 
## 
## Coefficients:
##          ar1      ar2      ma1
##       0.4594  -0.1664  -0.9394
## s.e.  0.0222   0.0219   0.0082
## 
## sigma^2 estimated as 85.19:  log likelihood=-7981.24
## AIC=15970.47   AICc=15970.49   BIC=15993.24
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set -0.01234418 9.223455 6.571991 NaN  Inf 0.6020914 0.004027249

##NOTE: AIC = 15970.47 , BIC = 15993.24 , RMSE = 9.223455

tsdisplay(residuals(model_Var9), lag.max = 45, main = '(1,1,1) Model Residuals')

#2.2 Auto Arima model with sesonality:

set.seed(301)
model2_Var9 <- auto.arima(train.sta_Var9, seasonal= TRUE)
summary(model2_Var9)
## Series: train.sta_Var9 
## ARIMA(5,0,3) with zero mean 
## 
## Coefficients:
##          ar1     ar2      ar3     ar4     ar5      ma1      ma2     ma3
##       0.2715  0.6444  -0.3462  0.1117  -0.019  -0.7471  -0.9168  0.6935
## s.e.  0.2485  0.1356   0.0760  0.0445   0.024   0.2478   0.0732  0.2245
## 
## sigma^2 estimated as 85.21:  log likelihood=-7978.98
## AIC=15975.95   AICc=15976.03   BIC=16027.18
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE         ACF1
## Training set -0.01021262 9.213898 6.567264 NaN  Inf 0.6016584 0.0006608765

##NOTE: AIC = 15975.95 , BIC = 16027.18 , RMSE = 9.213898

tsdisplay(residuals(model2_Var9), lag.max = 20, main = 'Seasonal Model Residuals')

NOTE: Above Graphs does not shows any serious lags.This indicates the good fit of the model.

Fit the observed with the predicted values for both models in the same plot.

par(mfrow = c(2,2))
fit_auto_train_Var9 %>% forecast(h=341) %>% autoplot() 

model2_Var9 %>% forecast(h=341) %>% autoplot() 

model_Var9 %>% forecast(h=341) %>% autoplot() 

COMMENT:
1. The plots includes 80% to 95% confidence interval.
2. For ARIMA models it plots the forecast but because of the nature of autoregressive process, these forecast stabilizes quite smoothly.

Plot and comment on the residuals of the fitted data for both models in the same plot.

Residual Plot - to confirm no problem with this model

par(mfrow= c(2,2))
hist(fit_auto_train_Var9$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(fit_auto_train_Var9$residuals))

hist(model_Var9$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model_Var9$residuals))

hist(model2_Var9$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model2_Var9$residuals))

OBSERVATION:
The residuals for each model are normally distributed, which shows the good fit of the models.However, on comparing all three models, residuals of “model_Var9” are normally distributed.

Testing the autocorrelation and partial autocorrelation of the residuals (using the plots and Ljung-Box Q statistic) up to an appropriate lag *

  1. ACF & PACF Plots
par(mfrow = c(3,2))
acf(fit_auto_train_Var9$residuals, main = 'Correlogram')
pacf(fit_auto_train_Var9$residuals, main = 'Partial Corelogram')

acf(model_Var9$residuals, main = 'Correlogram')
pacf(model_Var9$residuals, main = 'Partial Corelogram')

acf(model2_Var9$residuals, main = 'Correlogram')
pacf(model2_Var9$residuals, main = 'Partial Corelogram')

NOTE:
1. Residuals are the difference between actaual and fitted values
2. These blue lines are signficance bounds
3. This graph shows us the autocorelations for insample forecast errors
4. Do not exceed these significance bounds for lags 1 to end

#Ljunx-Box Test to check the autocorelations of the residuals

HO: No serial correlation upto 20 lags (does not exhibit lack of fit of the model)
Ha: Serial corelation is present (the model exhibits lack of fit)

We reject the null hypothesis, if p value is less than 0.05. This test is generally done for ARIMA models

Box.test(model_Var9$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model_Var9$residuals
## X-squared = 23.789, df = 20, p-value = 0.2517
Box.test(model2_Var9$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model2_Var9$residuals
## X-squared = 20.504, df = 20, p-value = 0.4268

OBSERVATIONS: For above performed Ljunx-Box Test on both the ARIMA models output, we can say that p-value is more than significance level 0.05, then there is not a statistical significance.
We conclude that there is a liitle evidence of non-zero auto-corelations in the insample forecast errors at lags 1 to 20.
So we fail to rejects the null hypothesis. Hence, we conclude that there is no serial auto corelation present between lags. Hence, the model seems to be good fit.

Test and comment on the normality of the residuals.

Here, further testing the normality by plotting Normal Q-Q plots. From below plots we observed that most of the datapoints lies on the line. Hence, we could conclude that our model produces residuals that are normal

par(mfrow = c(2,2))
qqnorm(fit_auto_train_Var9$residuals)
qqnorm(model_Var9$residuals)
qqnorm(model2_Var9$residuals)

#Validating The Model by testing the model performance with Holdout set

Here, choosing RMSE as evaluation parameter. The lower RMSE indicates a more accurate forecast.

Below, is the comparison of model. We are fitting the model to test set which is the holdout set. And checking the accuracy using RMSE(Root mean square error) evaluation parameter.

accuracy(forecast(fit_auto_train_Var9), test8) ["Test set", "RMSE"]
## [1] 11.11376

#o/p - 11.11376

accuracy(forecast(model2_Var9), test.sta_Var9) ["Test set", "RMSE"]
## [1] 9.527655

#o/p - 9.527655

accuracy(forecast(model_Var9), test.sta_Var9) ["Test set", "RMSE"]
## [1] 9.523064

#O/P - 9.523064

NOTE:- Here, the RMSE value is less for ARIMA models compared to exponential model.
The minimum RMSE is of model_Var9. Hence, it is forecasting better than other models in comparision.
However, The RMSE of auto arima models with seasonality and without seasonality have not much difference. It gives the same result for this series.

COMMENT ON BEST MODEL AND ITS PARAMETERS

We observed the best model to be ARIMA model i.e.“model_Var9”.

summary(model_Var9)
## Series: train.sta_Var9 
## ARIMA(2,0,1) with zero mean 
## 
## Coefficients:
##          ar1      ar2      ma1
##       0.4594  -0.1664  -0.9394
## s.e.  0.0222   0.0219   0.0082
## 
## sigma^2 estimated as 85.19:  log likelihood=-7981.24
## AIC=15970.47   AICc=15970.49   BIC=15993.24
## 
## Training set error measures:
##                       ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set -0.01234418 9.223455 6.571991 NaN  Inf 0.6020914 0.004027249

Passing parameter order = (2,0,1) ,which means

p = 2
d = 0
q = 1

For the best model, 2 members of lag are used as a predictor. As we transformed the series to stationary so members of differences needed for stationarity is 0. 1 is the number of lagged forecasts errors used in the prediction equation.

AIC = 15970.47, RMSE = 9.223455 which is minimum as compared to other models.

10. Forecasting on Sea level pressure (SLP)

#Checking the attributes of time series

attributes(tsSLP)
## $tsp
## [1] 1998.00 2004.94  365.00
## 
## $class
## [1] "ts"

#Plotting the series

plot(tsSLP, main = "Sea level pressure  (tsSLP)", col = "red")

From the plot, we observes there might be presence of some trend and seems to have seasonlity.

Checking the statistics and visualising the missing data in the series:

#Visualising the distribution of missing data

plotNA.distribution(tsSLP)

As we observes the data is missing at random, so locf and NOCB seems the good technique to implement here.

#Impute the missing values with “na_locf” function of “imputeTS package” and visualise the imputed values in the time series

tsSLP.imp <- na_locf(tsSLP)
plotNA.imputations(tsSLP, tsSLP.imp)

#Decomposition Of Time Series

decomp9 <- decompose(tsSLP.imp)
plot(decomp9, col = "red")

OBSERVATIONS:
1)There seems to have trend.
2)We observes upward trend from year 1998 to 2001 and downward trend from year 2001 to 2003.

3)We observe some seasonality in the series. Seasonality describes cyclical effects due to the time of the year.

Let’s explore this seasonality pattern by plotting the decomp$seasonal components such that y-axis represents Seasonality Index and x-axis has month

#Plotting the decomposed series

par(mfrow =c (1,2))
plot(decomp9$seasonal, type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")
plot(decomp9$seasonal[1:365], type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")

OBSERVATIONS:

  1. The plot shows a strong and significant sea level presuure in winter and summer.
  2. Also, no increase in the transition seasons.

Stationarity check of time series

We could check the stationarity of time series by ADF and KPSS Tests:

  1. Augmented Dickey- Fuller(ADF) t-statistic test for unit root
  2. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) for level or trend stationarity
adf.test(tsSLP.imp)
## Warning in adf.test(tsSLP.imp): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  tsSLP.imp
## Dickey-Fuller = -7.5884, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary

#p-vaue is smaller, hence it is stationary

kpss.test(tsSLP.imp, null="Trend")
## Warning in kpss.test(tsSLP.imp, null = "Trend"): p-value smaller than printed p-
## value
## 
##  KPSS Test for Trend Stationarity
## 
## data:  tsSLP.imp
## KPSS Trend = 0.2471, Truncation lag parameter = 8, p-value = 0.01

#p-vaue is smaller, hence it is not trend stationary

NOTE: From the both tests results, it leads to CASE4, we could recheck by visualising it i.e. by plotting ACF. Second, we can check each for characteristics of stationarity by looking at the autocorrelation functions (ACF) of each signal. For a stationary signal, because we expect no dependence with time, we would expect the ACF to go to 0 for each time lag (τ). Lets visualize the signals and ACFs.

Plotting ACF to check the auto-corelations

acf(tsSLP.imp,lag.max = length(tsSLP.imp),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Here, we observes most of the data lies outside the blue-dotted lines means the significance level. Now, we will take the difference to make our data stationary.

It seems difference of 1 is sufficient to make the series stationary.

tsSLP.sta <- diff(tsSLP.imp)
acf(tsSLP.sta, lag.max = length(tsSLP.sta),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

Hence, after differencing our series is now transformed to stationary time series.

#Visualising the stationary series

plot(tsSLP.sta, col = "red")

#CREATING MODELS AND FORECASTING: USING EXPONENTIAL SMOOTHING AND ARIMA TECHNIQUES FOR TIME SERIES FORECASTING

  1. TRAIN- TEST SET SPLIT
train9 <- window(tsSLP.imp, end = c(2004, 3))
test9 <- window(tsSLP.imp, start = c(2004, 4))

#Model 1 : Auto Exponential Smoothening Model

set.seed(301)
fit_auto_train_Var10 <- forecast(train9)
summary(fit_auto_train_Var10)
## 
## Forecast method: STL +  ETS(A,N,N)
## 
## Model Information:
## ETS(A,N,N) 
## 
## Call:
##  ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend) 
## 
##   Smoothing parameters:
##     alpha = 0.9999 
## 
##   Initial states:
##     l = 10261.7904 
## 
##   sigma:  31.7936
## 
##      AIC     AICc      BIC 
## 32047.14 32047.15 32064.22 
## 
## Error measures:
##                      ME     RMSE      MAE          MPE      MAPE      MASE
## Training set -0.1105768 31.77912 22.96737 -0.001582898 0.2258044 0.4703075
##                   ACF1
## Training set 0.1415532
## 
## Forecasts:
##           Point Forecast     Lo 80    Hi 80     Lo 95    Hi 95
## 2004.0082      10091.722 10050.977 10132.47 10029.408 10154.04
## 2004.0110      10075.243 10017.624 10132.86  9987.122 10163.36
## 2004.0137      10065.387  9994.819 10135.96  9957.463 10173.31
## 2004.0164      10029.654  9948.169 10111.14  9905.034 10154.27
## 2004.0192      10031.767  9940.665 10122.87  9892.439 10171.09
## 2004.0219      10055.272  9955.475 10155.07  9902.646 10207.90
## 2004.0247      10070.099  9962.307 10177.89  9905.245 10234.95
## 2004.0274      10041.672  9926.437 10156.91  9865.435 10217.91
## 2004.0301      10024.104  9901.880 10146.33  9837.178 10211.03
## 2004.0329      10039.122  9910.286 10167.96  9842.084 10236.16
## 2004.0356      10061.980  9926.856 10197.10  9855.326 10268.63
## 2004.0384      10055.219  9914.087 10196.35  9839.376 10271.06
## 2004.0411      10029.388  9882.493 10176.28  9804.731 10254.04
## 2004.0438      10009.440  9857.000 10161.88  9776.303 10242.58
## 2004.0466      10029.907  9872.117 10187.70  9788.587 10271.23
## 2004.0493      10032.100  9869.134 10195.06  9782.866 10281.33
## 2004.0521      10061.048  9893.067 10229.03  9804.143 10317.95
## 2004.0548      10098.535  9925.684 10271.39  9834.183 10362.89
## 2004.0575      10072.787  9895.200 10250.37  9801.191 10344.38
## 2004.0603      10084.969  9902.769 10267.17  9806.318 10363.62
## 2004.0630      10078.512  9891.812 10265.21  9792.979 10364.04
## 2004.0658      10036.213  9845.119 10227.31  9743.960 10328.46
## 2004.0685      10040.598  9845.210 10235.99  9741.778 10339.42
## 2004.0712      10069.321  9869.730 10268.91  9764.073 10374.57
## 2004.0740      10048.774  9845.068 10252.48  9737.232 10360.32
## 2004.0767      10037.117  9829.376 10244.86  9719.405 10354.83
## 2004.0795      10059.187  9847.489 10270.88  9735.423 10382.95
## 2004.0822      10036.210  9820.628 10251.79  9706.505 10365.91
## 2004.0849       9992.535  9773.137 10211.93  9656.994 10328.08
## 2004.0877      10017.232  9794.084 10240.38  9675.956 10358.51
## 2004.0904      10021.562  9794.725 10248.40  9674.644 10368.48
## 2004.0932      10028.305  9797.838 10258.77  9675.836 10380.77
## 2004.0959      10042.778  9808.738 10276.82  9684.844 10400.71
## 2004.0986      10045.776  9808.216 10283.34  9682.459 10409.09
## 2004.1014      10050.357  9809.329 10291.39  9681.737 10418.98
## 2004.1041      10058.648  9814.201 10303.10  9684.798 10432.50
## 2004.1068      10055.265  9807.445 10303.08  9676.258 10434.27
## 2004.1096      10043.019  9791.874 10294.16  9658.925 10427.11
## 2004.1123      10029.918  9775.489 10284.35  9640.803 10419.03
## 2004.1151       9999.729  9742.059 10257.40  9605.657 10393.80
## 2004.1178       9995.304  9734.433 10256.17  9596.337 10394.27
## 2004.1205      10026.522  9762.488 10290.55  9622.718 10430.33
## 2004.1233      10054.495  9787.337 10321.65  9645.912 10463.08
## 2004.1260      10034.255  9764.008 10304.50  9620.948 10447.56
## 2004.1288      10026.067  9752.767 10299.37  9608.090 10444.04
## 2004.1315       9995.194  9718.874 10271.51  9572.599 10417.79
## 2004.1342       9997.537  9718.230 10276.84  9570.373 10424.70
## 2004.1370      10028.914  9746.651 10311.18  9597.230 10460.60
## 2004.1397      10055.795  9770.607 10340.98  9619.638 10491.95
## 2004.1425      10036.588  9748.504 10324.67  9596.002 10477.17
## 2004.1452      10024.230  9733.280 10315.18  9579.261 10469.20
## 2004.1479      10004.319  9710.530 10298.11  9555.008 10453.63
## 2004.1507       9975.967  9679.367 10272.57  9522.356 10429.58
## 2004.1534       9993.885  9694.500 10293.27  9536.015 10451.75
## 2004.1562       9996.275  9694.131 10298.42  9534.185 10458.37
## 2004.1589       9985.688  9680.810 10290.57  9519.416 10451.96
## 2004.1616      10006.494  9698.905 10314.08  9536.078 10476.91
## 2004.1644      10019.232  9708.956 10329.51  9544.706 10493.76
## 2004.1671       9979.492  9666.553 10292.43  9500.893 10458.09
## 2004.1699       9979.662  9664.082 10295.24  9497.024 10462.30
## 2004.1726       9999.190  9680.991 10317.39  9512.547 10485.83
## 2004.1753      10003.270  9682.474 10324.07  9512.655 10493.89
## 2004.1781       9973.023  9649.650 10296.40  9478.466 10467.58
## 2004.1808       9975.146  9649.217 10301.08  9476.681 10473.61
## 2004.1836      10014.936  9686.470 10343.40  9512.591 10517.28
## 2004.1863      10080.233  9749.251 10411.22  9574.039 10586.43
## 2004.1890      10081.442  9747.961 10414.92  9571.427 10591.46
## 2004.1918      10050.193  9714.233 10386.15  9536.386 10564.00
## 2004.1945      10037.130  9698.709 10375.55  9519.559 10554.70
## 2004.1973      10034.273  9693.408 10375.14  9512.965 10555.58
## 2004.2000      10031.458  9688.167 10374.75  9506.440 10556.48
## 2004.2027       9998.990  9653.290 10344.69  9470.288 10527.69
## 2004.2055       9953.814  9605.722 10301.91  9421.453 10486.18
## 2004.2082       9961.241  9610.773 10311.71  9425.246 10497.24
## 2004.2110       9988.131  9635.302 10340.96  9448.526 10527.74
## 2004.2137      10008.162  9652.989 10363.33  9464.971 10551.35
## 2004.2164       9997.975  9640.473 10355.48  9451.222 10544.73
## 2004.2192       9999.669  9639.853 10359.48  9449.378 10549.96
## 2004.2219      10037.442  9675.327 10399.56  9483.635 10591.25
## 2004.2247      10057.870  9693.470 10422.27  9500.569 10615.17
## 2004.2274      10053.072  9686.402 10419.74  9492.299 10613.85
## 2004.2301      10031.290  9662.363 10400.22  9467.065 10595.51
## 2004.2329       9994.823  9623.653 10365.99  9427.168 10562.48
## 2004.2356      10008.424  9635.025 10381.82  9437.360 10579.49
## 2004.2384      10013.222  9637.607 10388.84  9438.769 10587.67
## 2004.2411       9993.509  9615.691 10371.33  9415.686 10571.33
## 2004.2438       9997.599  9617.591 10377.61  9416.427 10578.77
## 2004.2466      10007.679  9625.494 10389.87  9423.177 10592.18
## 2004.2493      10016.287  9631.936 10400.64  9428.473 10604.10
## 2004.2521       9986.104  9599.600 10372.61  9394.997 10577.21
## 2004.2548       9974.317  9585.672 10362.96  9379.935 10568.70
## 2004.2575       9994.935  9604.160 10385.71  9397.296 10592.57
## 2004.2603      10024.823  9631.929 10417.72  9423.944 10625.70
## 2004.2630      10022.726  9627.726 10417.73  9418.626 10626.83
## 2004.2658      10011.593  9614.498 10408.69  9404.288 10618.90
## 2004.2685       9991.809  9592.629 10390.99  9381.316 10602.30
## 2004.2712       9992.280  9591.026 10393.53  9378.616 10605.94
## 2004.2740      10003.998  9600.681 10407.31  9387.178 10620.82
## 2004.2767      10000.009  9594.640 10405.38  9380.051 10619.97
## 2004.2795      10026.569  9619.157 10433.98  9403.487 10649.65
## 2004.2822      10023.101  9613.658 10432.54  9396.912 10649.29
## 2004.2849      10010.385  9598.920 10421.85  9381.103 10639.67
## 2004.2877       9996.607  9583.129 10410.08  9364.248 10628.97
## 2004.2904       9996.140  9580.660 10411.62  9360.719 10631.56
## 2004.2932      10001.518  9584.046 10418.99  9363.049 10639.99
## 2004.2959       9989.366  9569.910 10408.82  9347.864 10630.87
## 2004.2986       9981.059  9559.630 10402.49  9336.538 10625.58
## 2004.3014       9989.553  9566.159 10412.95  9342.027 10637.08
## 2004.3041      10003.444  9578.095 10428.79  9352.928 10653.96
## 2004.3068      10007.282  9579.986 10434.58  9353.789 10660.78
## 2004.3096       9994.869  9565.635 10424.10  9338.412 10651.33
## 2004.3123      10000.600  9569.436 10431.76  9341.192 10660.01
## 2004.3151      10001.673  9568.589 10434.76  9339.328 10664.02
## 2004.3178       9991.247  9556.251 10426.24  9325.978 10656.52
## 2004.3205       9993.830  9556.930 10430.73  9325.650 10662.01
## 2004.3233       9995.779  9556.983 10434.57  9324.699 10666.86
## 2004.3260       9990.785  9550.102 10431.47  9316.819 10664.75
## 2004.3288       9999.122  9556.560 10441.68  9322.282 10675.96
## 2004.3315      10022.692  9578.258 10467.12  9342.990 10702.39
## 2004.3342      10016.741  9570.445 10463.04  9334.189 10699.29
## 2004.3370       9997.129  9548.976 10445.28  9311.739 10682.52
## 2004.3397       9975.876  9525.876 10425.88  9287.660 10664.09
## 2004.3425       9972.995  9521.154 10424.84  9281.964 10664.03
## 2004.3452      10008.798  9555.124 10462.47  9314.964 10702.63
## 2004.3479       9998.552  9543.053 10454.05  9301.926 10695.18
## 2004.3507       9981.076  9523.758 10438.39  9281.669 10680.48
## 2004.3534       9985.707  9526.578 10444.84  9283.530 10687.88
## 2004.3562       9993.544  9532.611 10454.48  9288.607 10698.48
## 2004.3589       9975.344  9512.614 10438.07  9267.660 10683.03
## 2004.3616       9970.452  9505.932 10434.97  9260.030 10680.87
## 2004.3644       9981.491  9515.187 10447.79  9268.341 10694.64
## 2004.3671       9987.355  9519.275 10455.43  9271.488 10703.22
## 2004.3699       9981.449  9511.600 10451.30  9262.876 10700.02
## 2004.3726       9977.285  9505.672 10448.90  9256.015 10698.55
## 2004.3753       9976.672  9503.303 10450.04  9252.717 10700.63
## 2004.3781       9980.080  9504.961 10455.20  9253.448 10706.71
## 2004.3808       9991.751  9514.888 10468.61  9262.453 10721.05
## 2004.3836       9984.588  9505.989 10463.19  9252.633 10716.54
## 2004.3863       9978.438  9498.107 10458.77  9243.835 10713.04
## 2004.3890       9971.923  9489.867 10453.98  9234.683 10709.16
## 2004.3918       9980.646  9496.872 10464.42  9240.778 10720.51
## 2004.3945       9984.189  9498.703 10469.68  9241.702 10726.68
## 2004.3973       9985.440  9498.247 10472.63  9240.343 10730.54
## 2004.4000       9984.413  9495.520 10473.31  9236.716 10732.11
## 2004.4027       9982.578  9491.990 10473.17  9232.288 10732.87
## 2004.4055       9975.932  9483.655 10468.21  9223.059 10728.80
## 2004.4082       9976.285  9482.325 10470.24  9220.839 10731.73
## 2004.4110       9981.684  9486.047 10477.32  9223.673 10739.70
## 2004.4137       9979.056  9481.748 10476.37  9218.488 10739.62
## 2004.4164       9975.768  9476.793 10474.74  9212.652 10738.88
## 2004.4192       9987.543  9486.907 10488.18  9221.887 10753.20
## 2004.4219       9992.549  9490.259 10494.84  9224.363 10760.74
## 2004.4247       9991.107  9487.168 10495.05  9220.398 10761.82
## 2004.4274       9988.917  9483.333 10494.50  9215.693 10762.14
## 2004.4301      10005.768  9498.545 10512.99  9230.038 10781.50
## 2004.4329      10021.061  9512.205 10529.92  9242.833 10799.29
## 2004.4356      10026.901  9516.416 10537.39  9246.182 10807.62
## 2004.4384      10000.499  9488.391 10512.61  9217.297 10783.70
## 2004.4411       9993.378  9479.652 10507.10  9207.702 10779.05
## 2004.4438       9995.309  9479.970 10510.65  9207.166 10783.45
## 2004.4466      10007.782  9490.835 10524.73  9217.180 10798.38
## 2004.4493      10012.734  9494.185 10531.28  9219.681 10805.79
## 2004.4521      10017.521  9497.373 10537.67  9222.024 10813.02
## 2004.4548      10015.338  9493.597 10537.08  9217.404 10813.27
## 2004.4575      10022.710  9499.380 10546.04  9222.347 10823.07
## 2004.4603      10023.532  9498.620 10548.44  9220.748 10826.32
## 2004.4630      10016.269  9489.777 10542.76  9211.070 10821.47
## 2004.4658      10011.425  9483.360 10539.49  9203.819 10819.03
## 2004.4685      10011.596  9481.962 10541.23  9201.590 10821.60
## 2004.4712      10011.038  9479.839 10542.24  9198.640 10823.44
## 2004.4740      10007.009  9474.250 10539.77  9192.224 10821.79
## 2004.4767      10006.301  9471.986 10540.62  9189.137 10823.46
## 2004.4795      10017.119  9481.253 10552.98  9197.583 10836.65
## 2004.4822      10026.781  9489.369 10564.19  9204.881 10848.68
## 2004.4849      10018.937  9479.983 10557.89  9194.677 10843.20
## 2004.4877      10010.590  9470.098 10551.08  9183.979 10837.20
## 2004.4904      10004.486  9462.461 10546.51  9175.530 10833.44
## 2004.4932      10001.377  9457.823 10544.93  9170.083 10832.67
## 2004.4959      10001.496  9456.417 10546.57  9167.870 10835.12
## 2004.4986      10002.880  9456.280 10549.48  9166.928 10838.83
## 2004.5014      10004.290  9456.174 10552.41  9166.020 10842.56
## 2004.5041      10004.582  9454.954 10554.21  9163.999 10845.16
## 2004.5068      10008.735  9457.599 10559.87  9165.846 10851.62
## 2004.5096      10024.934  9472.295 10577.57  9179.746 10870.12
## 2004.5123      10020.065  9465.926 10574.20  9172.583 10867.55
## 2004.5151      10005.839  9450.205 10561.47  9156.069 10855.61
## 2004.5178      10000.523  9443.397 10557.65  9148.472 10852.57
## 2004.5205      10000.422  9441.808 10559.04  9146.096 10854.75
## 2004.5233      10003.214  9443.116 10563.31  9146.619 10859.81
## 2004.5260       9997.079  9435.501 10558.66  9138.220 10855.94
## 2004.5288       9990.041  9426.987 10553.09  9128.925 10851.16
## 2004.5315       9994.288  9429.763 10558.81  9130.921 10857.66
## 2004.5342      10003.981  9437.987 10569.97  9138.369 10869.59
## 2004.5370      10014.152  9446.694 10581.61  9146.300 10882.00
## 2004.5397      10004.604  9435.686 10573.52  9134.519 10874.69
## 2004.5425       9993.205  9422.830 10563.58  9120.891 10865.52
## 2004.5452       9990.594  9418.765 10562.42  9116.057 10865.13
## 2004.5479       9998.378  9425.100 10571.66  9121.625 10875.13
## 2004.5507      10011.893  9437.169 10586.62  9132.928 10890.86
## 2004.5534      10003.111  9426.945 10579.28  9121.941 10884.28
## 2004.5562       9995.095  9417.491 10572.70  9111.725 10878.47
## 2004.5589      10002.943  9423.903 10581.98  9117.378 10888.51
## 2004.5616      10009.816  9429.344 10590.29  9122.061 10897.57
## 2004.5644      10010.432  9428.533 10592.33  9120.494 10900.37
## 2004.5671      10008.449  9425.126 10591.77  9116.333 10900.57
## 2004.5699      10011.653  9426.908 10596.40  9117.363 10905.94
## 2004.5726      10014.235  9428.072 10600.40  9117.777 10910.69
## 2004.5753      10013.402  9425.825 10600.98  9114.781 10912.02
## 2004.5781      10014.407  9425.420 10603.39  9113.629 10915.19
## 2004.5808      10006.270  9415.875 10596.66  9103.339 10909.20
## 2004.5836       9997.285  9405.487 10589.08  9092.207 10902.36
## 2004.5863      10006.399  9413.200 10599.60  9099.179 10913.62
## 2004.5890      10017.764  9423.167 10612.36  9108.407 10927.12
## 2004.5918      10019.154  9423.163 10615.15  9107.665 10930.64
## 2004.5945      10012.502  9415.120 10609.88  9098.885 10926.12
## 2004.5973      10002.450  9403.680 10601.22  9086.711 10918.19
## 2004.6000       9998.981  9398.827 10599.14  9081.125 10916.84
## 2004.6027       9999.372  9397.836 10600.91  9079.403 10919.34
## 2004.6055      10000.718  9397.805 10603.63  9078.642 10922.79
## 2004.6082      10001.000  9396.712 10605.29  9076.821 10925.18
## 2004.6110      10006.234  9400.574 10611.89  9079.957 10932.51
## 2004.6137      10006.487  9399.458 10613.52  9078.116 10934.86
## 2004.6164       9996.248  9387.854 10604.64  9065.789 10926.71
## 2004.6192       9985.690  9375.933 10595.45  9053.147 10918.23
## 2004.6219       9994.671  9383.554 10605.79  9060.048 10929.29
## 2004.6247       9993.909  9381.436 10606.38  9057.212 10930.61
## 2004.6274       9983.652  9369.825 10597.48  9044.885 10922.42
## 2004.6301       9969.655  9354.478 10584.83  9028.823 10910.49
## 2004.6329       9976.776  9360.251 10593.30  9033.882 10919.67
## 2004.6356       9981.599  9363.730 10599.47  9036.649 10926.55
## 2004.6384       9965.202  9345.991 10584.41  9018.200 10912.20
## 2004.6411       9964.504  9343.954 10585.05  9015.455 10913.55
## 2004.6438       9980.673  9358.787 10602.56  9029.580 10931.77
## 2004.6466       9988.298  9365.079 10611.52  9035.167 10941.43
## 2004.6493       9979.885  9355.336 10604.43  9024.720 10935.05
## 2004.6521       9976.530  9350.653 10602.41  9019.334 10933.73
## 2004.6548       9978.180  9350.979 10605.38  9018.959 10937.40
## 2004.6575       9991.070  9362.547 10619.59  9029.827 10952.31
## 2004.6603       9994.148  9364.305 10623.99  9030.887 10957.41
## 2004.6630       9982.251  9351.092 10613.41  9016.977 10947.52
## 2004.6658       9973.850  9341.378 10606.32  9006.568 10941.13
## 2004.6685       9989.636  9355.854 10623.42  9020.349 10958.92
## 2004.6712       9983.209  9348.118 10618.30  9011.921 10954.50
## 2004.6740       9981.434  9345.037 10617.83  9008.149 10954.72
## 2004.6767       9989.096  9351.396 10626.79  9013.819 10964.37
## 2004.6795       9975.739  9336.740 10614.74  8998.474 10953.00
## 2004.6822       9952.052  9311.755 10592.35  8972.802 10931.30
## 2004.6849       9966.710  9325.119 10608.30  8985.481 10947.94
## 2004.6877       9992.065  9349.182 10634.95  9008.860 10975.27
## 2004.6904      10003.955  9359.781 10648.13  9018.777 10989.13
## 2004.6932      10003.516  9358.055 10648.98  9016.369 10990.66
## 2004.6959       9999.879  9353.134 10646.62  9010.768 10988.99
## 2004.6986       9995.315  9347.288 10643.34  9004.243 10986.39
## 2004.7014       9990.310  9341.004 10639.62  8997.282 10983.34
## 2004.7041       9987.340  9336.756 10637.92  8992.358 10982.32
## 2004.7068      10004.192  9352.335 10656.05  9007.262 11001.12
## 2004.7096      10007.390  9354.260 10660.52  9008.514 11006.26
## 2004.7123      10011.434  9357.035 10665.83  9010.617 11012.25
## 2004.7151       9995.684  9340.018 10651.35  8992.930 10998.44
## 2004.7178       9995.321  9338.391 10652.25  8990.633 11000.01
## 2004.7205      10010.256  9352.063 10668.45  9003.637 11016.87
## 2004.7233      10009.830  9350.377 10669.28  9001.284 11018.37
## 2004.7260      10005.418  9344.708 10666.13  8994.949 11015.89
## 2004.7288       9996.196  9334.232 10658.16  8983.809 11008.58
## 2004.7315      10020.831  9357.614 10684.05  9006.528 11035.13
## 2004.7342      10033.839  9369.372 10698.31  9017.624 11050.05
## 2004.7370      10048.582  9382.867 10714.30  9030.459 11066.71
## 2004.7397      10037.252  9370.291 10704.21  9017.223 11057.28
## 2004.7425      10028.806  9360.602 10697.01  9006.877 11050.74
## 2004.7452      10023.244  9353.799 10692.69  8999.417 11047.07
## 2004.7479      10000.145  9329.462 10670.83  8974.424 11025.87
## 2004.7507      10002.818  9330.899 10674.74  8975.206 11030.43
## 2004.7534      10013.159  9340.006 10686.31  8983.660 11042.66
## 2004.7562      10019.619  9345.233 10694.00  8988.236 11051.00
## 2004.7589      10028.618  9353.004 10704.23  8995.355 11061.88
## 2004.7616      10028.287  9351.445 10705.13  8993.146 11063.43
## 2004.7644      10010.075  9332.008 10688.14  8973.061 11047.09
## 2004.7671      10005.387  9326.097 10684.68  8966.502 11044.27
## 2004.7699      10009.607  9329.097 10690.12  8968.856 11050.36
## 2004.7726      10022.912  9341.183 10704.64  8980.297 11065.53
## 2004.7753      10034.756  9351.811 10717.70  8990.282 11079.23
## 2004.7781      10044.769  9360.610 10728.93  8998.438 11091.10
## 2004.7808      10038.720  9353.349 10724.09  8990.535 11086.91
## 2004.7836      10037.199  9350.618 10723.78  8987.164 11087.23
## 2004.7863      10033.306  9345.518 10721.10  8981.424 11085.19
## 2004.7890      10061.768  9372.774 10750.76  9008.042 11115.49
## 2004.7918      10085.104  9394.906 10775.30  9029.537 11140.67
## 2004.7945      10083.046  9391.647 10774.45  9025.642 11140.45
## 2004.7973      10077.970  9385.371 10770.57  9018.732 11137.21
## 2004.8000      10060.887  9367.091 10754.68  8999.818 11121.96
## 2004.8027      10047.291  9352.300 10742.28  8984.394 11110.19
## 2004.8055      10063.800  9367.615 10759.98  8999.078 11128.52
## 2004.8082      10064.381  9367.006 10761.76  8997.838 11130.92
## 2004.8110      10067.327  9368.763 10765.89  8998.965 11135.69
## 2004.8137      10020.251  9320.500 10720.00  8950.074 11090.43
## 2004.8164      10027.278  9326.342 10728.21  8955.288 11099.27
## 2004.8192      10030.949  9328.829 10733.07  8957.150 11104.75
## 2004.8219      10024.192  9320.891 10727.49  8948.587 11099.80
## 2004.8247      10056.264  9351.785 10760.74  8978.856 11133.67
## 2004.8274      10044.581  9338.925 10750.24  8965.373 11123.79
## 2004.8301      10038.023  9331.191 10744.85  8957.017 11119.03
## 2004.8329      10027.330  9319.325 10735.33  8944.530 11110.13
## 2004.8356      10062.992  9353.816 10772.17  8978.401 11147.58
## 2004.8384      10064.429  9354.083 10774.77  8978.049 11150.81
## 2004.8411      10026.766  9315.254 10738.28  8938.602 11114.93
## 2004.8438      10035.869  9323.191 10748.55  8945.922 11125.82
## 2004.8466      10069.399  9355.557 10783.24  8977.672 11161.13
## 2004.8493      10093.408  9378.405 10808.41  8999.905 11186.91
## 2004.8521      10091.462  9375.299 10807.63  8996.185 11186.74
## 2004.8548      10070.571  9353.250 10787.89  8973.523 11167.62
## 2004.8575      10068.548  9350.070 10787.02  8969.732 11167.36
## 2004.8603      10030.662  9311.031 10750.29  8930.081 11131.24
## 2004.8630       9990.480  9269.697 10711.26  8888.137 11092.82
## 2004.8658      10010.923  9288.989 10732.86  8906.820 11115.03
## 2004.8685      10043.698  9320.615 10766.78  8937.839 11149.56
## 2004.8712      10052.981  9328.751 10777.21  8945.367 11160.59
## 2004.8740      10048.018  9322.643 10773.39  8938.653 11157.38
## 2004.8767      10037.934  9311.416 10764.45  8926.821 11149.05
## 2004.8795      10062.520  9334.861 10790.18  8949.662 11175.38
## 2004.8822      10067.274  9338.475 10796.07  8952.673 11181.88
## 2004.8849      10062.958  9333.021 10792.89  8946.616 11179.30
## 2004.8877      10049.891  9318.818 10780.96  8931.812 11167.97
## 2004.8904      10046.852  9314.645 10779.06  8927.038 11166.67
## 2004.8932      10063.268  9329.928 10796.61  8941.721 11184.81
## 2004.8959      10054.851  9320.381 10789.32  8931.575 11178.13
## 2004.8986      10061.361  9325.761 10796.96  8936.358 11186.36
## 2004.9014      10039.769  9303.042 10776.50  8913.042 11166.50
## 2004.9041      10046.557  9308.705 10784.41  8918.109 11175.01
## 2004.9068      10037.986  9299.010 10776.96  8907.819 11168.15
## 2004.9096      10045.275  9305.176 10785.37  8913.391 11177.16
## 2004.9123      10030.917  9289.698 10772.14  8897.320 11164.51
## 2004.9151      10001.777  9259.439 10744.12  8866.469 11137.09
## 2004.9178      10022.099  9278.644 10765.55  8885.083 11159.12
## 2004.9205      10029.009  9284.438 10773.58  8890.286 11167.73
## 2004.9233      10047.419  9301.734 10793.10  8906.993 11187.85
## 2004.9260      10065.511  9318.714 10812.31  8923.384 11207.64
## 2004.9288      10066.752  9318.845 10814.66  8922.927 11210.58
## 2004.9315      10049.229  9300.213 10798.24  8903.708 11194.75
## 2004.9342      10082.606  9332.483 10832.73  8935.392 11229.82
## 2004.9370      10064.194  9312.965 10815.42  8915.289 11213.10
## 2004.9397      10064.438  9312.105 10816.77  8913.844 11215.03
## 2004.9425      10056.260  9302.825 10809.69  8903.980 11208.54
## 2004.9452      10069.058  9314.522 10823.59  8915.095 11223.02
## 2004.9479      10070.393  9314.758 10826.03  8914.749 11226.04
## 2004.9507      10066.078  9309.346 10822.81  8908.756 11223.40
## 2004.9534      10071.786  9313.958 10829.61  8912.788 11230.78
## 2004.9562      10066.063  9307.140 10824.99  8905.391 11226.73
## 2004.9589      10073.946  9313.930 10833.96  8911.602 11236.29
## 2004.9616      10073.379  9312.273 10834.49  8909.367 11237.39
## 2004.9644      10073.981  9311.785 10836.18  8908.302 11239.66
## 2004.9671      10077.150  9313.866 10840.43  8909.808 11244.49
## 2004.9699      10105.396  9341.026 10869.77  8936.392 11274.40
## 2004.9726      10117.362  9351.907 10882.82  8946.699 11288.03
## 2004.9753      10104.318  9337.778 10870.86  8931.997 11276.64
## 2004.9781      10079.751  9312.130 10847.37  8905.776 11253.73
## 2004.9808      10054.534  9285.833 10823.24  8878.907 11230.16
## 2004.9836      10039.199  9269.418 10808.98  8861.921 11216.48
## 2004.9863      10048.477  9277.619 10819.33  8869.551 11227.40
## 2004.9890      10052.697  9280.763 10824.63  8872.126 11233.27
## 2004.9918      10066.834  9293.826 10839.84  8884.620 11249.05
## 2004.9945      10069.695  9295.614 10843.78  8885.840 11253.55
## 2004.9973      10087.467  9312.315 10862.62  8901.974 11272.96
## 2005.0000      10064.366  9288.144 10840.59  8877.237 11251.50
## 2005.0027      10067.201  9289.910 10844.49  8878.438 11255.96
## 2005.0055      10075.013  9296.655 10853.37  8884.618 11265.41
## 2005.0082      10091.722  9312.299 10871.15  8899.698 11283.75
## 2005.0110      10075.243  9294.756 10855.73  8881.591 11268.90
## 2005.0137      10065.387  9283.838 10846.94  8870.110 11260.66
## 2005.0164      10029.654  9247.043 10812.26  8832.754 11226.55
## 2005.0192      10031.767  9248.096 10815.44  8833.246 11230.29
## 2005.0219      10055.272  9270.543 10840.00  8855.132 11255.41
## 2005.0247      10070.099  9284.314 10855.89  8868.344 11271.86
## 2005.0274      10041.672  9254.830 10828.51  8838.302 11245.04
## 2005.0301      10024.104  9236.209 10812.00  8819.122 11229.09
## 2005.0329      10039.122  9250.174 10828.07  8832.530 11245.71
## 2005.0356      10061.980  9271.981 10851.98  8853.781 11270.18
## 2005.0384      10055.219  9264.170 10846.27  8845.414 11265.02
## 2005.0411      10029.388  9237.291 10821.49  8817.979 11240.80
## 2005.0438      10009.440  9216.296 10802.58  8796.430 11222.45
## 2005.0466      10029.907  9235.717 10824.10  8815.298 11244.52
## 2005.0493      10032.100  9236.865 10827.33  8815.893 11248.31
## 2005.0521      10061.048  9264.770 10857.33  8843.247 11278.85
## 2005.0548      10098.535  9301.216 10895.85  8879.141 11317.93
## 2005.0575      10072.787  9274.428 10871.15  8851.802 11293.77
## 2005.0603      10084.969  9285.571 10884.37  8862.395 11307.54
## 2005.0630      10078.512  9278.076 10878.95  8854.351 11302.67
## 2005.0658      10036.213  9234.741 10837.68  8810.467 11261.96
## 2005.0685      10040.598  9238.092 10843.10  8813.271 11267.93
## 2005.0712      10069.321  9265.781 10872.86  8840.412 11298.23
## 2005.0740      10048.774  9244.202 10853.35  8818.287 11279.26
## 2005.0767      10037.117  9231.514 10842.72  8805.053 11269.18
## 2005.0795      10059.187  9252.554 10865.82  8825.549 11292.82
## 2005.0822      10036.210  9228.549 10843.87  8801.000 11271.42
## 2005.0849       9992.535  9183.847 10801.22  8755.754 11229.32
## 2005.0877      10017.232  9207.519 10826.95  8778.883 11255.58
## 2005.0904      10021.562  9210.825 10832.30  8781.646 11261.48
## 2005.0932      10028.305  9216.544 10840.07  8786.824 11269.79
## 2005.0959      10042.778  9229.996 10855.56  8799.735 11285.82
## 2005.0986      10045.776  9231.973 10859.58  8801.172 11290.38
## 2005.1014      10050.357  9235.536 10865.18  8804.195 11296.52
## 2005.1041      10058.648  9242.808 10874.49  8810.929 11306.37
## 2005.1068      10055.265  9238.408 10872.12  8805.990 11304.54
## 2005.1096      10043.019  9225.147 10860.89  8792.192 11293.85
## 2005.1123      10029.918  9211.032 10848.80  8777.540 11282.30
## 2005.1151       9999.729  9179.830 10819.63  8745.802 11253.66
## 2005.1178       9995.304  9174.394 10816.21  8739.830 11250.78
## 2005.1205      10026.522  9204.601 10848.44  8769.502 11283.54
## 2005.1233      10054.495  9231.565 10877.42  8795.932 11313.06
## 2005.1260      10034.255  9210.317 10858.19  8774.151 11294.36
## 2005.1288      10026.067  9201.122 10851.01  8764.423 11287.71
## 2005.1315       9995.194  9169.245 10821.14  8732.013 11258.38
## 2005.1342       9997.537  9170.583 10824.49  8732.820 11262.25
## 2005.1370      10028.914  9200.957 10856.87  8762.663 11295.16
## 2005.1397      10055.795  9226.837 10884.75  8788.013 11323.58
## 2005.1425      10036.588  9206.629 10866.55  8767.275 11305.90
## 2005.1452      10024.230  9193.272 10855.19  8753.389 11295.07
## 2005.1479      10004.319  9172.362 10836.28  8731.951 11276.69
## 2005.1507       9975.967  9143.013 10808.92  8702.074 11249.86
## 2005.1534       9993.885  9159.935 10827.83  8718.469 11269.30
## 2005.1562       9996.275  9161.331 10831.22  8719.339 11273.21
## 2005.1589       9985.688  9149.751 10821.63  8707.233 11264.14
## 2005.1616      10006.494  9169.565 10843.42  8726.521 11286.47
## 2005.1644      10019.232  9181.311 10857.15  8737.743 11300.72
## 2005.1671       9979.492  9140.581 10818.40  8696.489 11262.49
## 2005.1699       9979.662  9139.762 10819.56  8695.147 11264.18
## 2005.1726       9999.190  9158.303 10840.08  8713.164 11285.21
## 2005.1753      10003.270  9161.397 10845.14  8715.737 11290.80
## 2005.1781       9973.023  9130.164 10815.88  8683.982 11262.06
## 2005.1808       9975.146  9131.304 10818.99  8684.601 11265.69
## 2005.1836      10014.936  9170.110 10859.76  8722.887 11306.98
## 2005.1863      10080.233  9234.426 10926.04  8786.683 11373.78
## 2005.1890      10081.442  9234.654 10928.23  8786.392 11376.49
## 2005.1918      10050.193  9202.426 10897.96  8753.645 11346.74
## 2005.1945      10037.130  9188.384 10885.88  8739.085 11335.17
## 2005.1973      10034.273  9184.550 10884.00  8734.734 11333.81
## 2005.2000      10031.458  9180.759 10882.16  8730.427 11332.49
## 2005.2027       9998.990  9147.316 10850.66  8696.467 11301.51
## 2005.2055       9953.814  9101.166 10806.46  8649.802 11257.83
## 2005.2082       9961.241  9107.620 10814.86  8655.741 11266.74
## 2005.2110       9988.131  9133.538 10842.72  8681.144 11295.12
## 2005.2137      10008.162  9152.599 10863.72  8699.691 11316.63
## 2005.2164       9997.975  9141.442 10854.51  8688.021 11307.93
## 2005.2192       9999.669  9142.168 10857.17  8688.234 11311.10
## 2005.2219      10037.442  9178.974 10895.91  8724.529 11350.36
## 2005.2247      10057.870  9198.436 10917.30  8743.479 11372.26
## 2005.2274      10053.072  9192.673 10913.47  8737.205 11368.94
## 2005.2301      10031.290  9169.926 10892.65  8713.948 11348.63
## 2005.2329       9994.823  9132.496 10857.15  8676.008 11313.64
## 2005.2356      10008.424  9145.135 10871.71  8688.138 11328.71
## 2005.2384      10013.222  9148.973 10877.47  8691.467 11334.98
## 2005.2411       9993.509  9128.300 10858.72  8670.286 11316.73
## 2005.2438       9997.599  9131.431 10863.77  8672.910 11322.29
## 2005.2466      10007.679  9140.554 10874.80  8681.526 11333.83
## 2005.2493      10016.287  9148.205 10884.37  8688.671 11343.90
## 2005.2521       9986.104  9117.067 10855.14  8657.026 11315.18
## 2005.2548       9974.317  9104.326 10844.31  8643.780 11304.85
## 2005.2575       9994.935  9123.990 10865.88  8662.940 11326.93
## 2005.2603      10024.823  9152.925 10896.72  8691.370 11358.27
## 2005.2630      10022.726  9149.878 10895.58  8687.819 11357.63
## 2005.2658      10011.593  9137.794 10885.39  8675.233 11347.95
## 2005.2685       9991.809  9117.061 10866.56  8653.997 11329.62
## 2005.2712       9992.280  9116.583 10867.98  8653.017 11331.54
## 2005.2740      10003.998  9127.354 10880.64  8663.286 11344.71
## 2005.2767      10000.009  9122.419 10877.60  8657.851 11342.17
## 2005.2795      10026.569  9148.033 10905.10  8682.965 11370.17
## 2005.2822      10023.101  9143.622 10902.58  8678.054 11368.15
## 2005.2849      10010.385  9129.963 10890.81  8663.895 11356.88
## 2005.2877       9996.607  9115.242 10877.97  8648.676 11344.54
## 2005.2904       9996.140  9113.834 10878.45  8646.770 11345.51
## 2005.2932      10001.518  9118.272 10884.76  8650.710 11352.33
## 2005.2959       9989.366  9105.181 10873.55  8637.122 11341.61
## 2005.2986       9981.059  9095.936 10866.18  8627.380 11334.74
## 2005.3014       9989.553  9103.493 10875.61  8634.441 11344.67
## 2005.3041      10003.444  9116.448 10890.44  8646.900 11359.99
## 2005.3068      10007.282  9119.351 10895.21  8649.308 11365.26
## 2005.3096       9994.869  9106.003 10883.73  8635.466 11354.27
## 2005.3123      10000.600  9110.801 10890.40  8639.770 11361.43
## 2005.3151      10001.673  9110.942 10892.40  8639.417 11363.93
## 2005.3178       9991.247  9099.585 10882.91  8627.567 11354.93
## 2005.3205       9993.830  9101.238 10886.42  8628.728 11358.93
## 2005.3233       9995.779  9102.257 10889.30  8629.255 11362.30
## 2005.3260       9990.785  9096.334 10885.23  8622.841 11358.73
## 2005.3288       9999.122  9103.745 10894.50  8629.760 11368.48
## 2005.3315      10022.692  9126.388 10919.00  8651.913 11393.47
## 2005.3342      10016.741  9119.512 10913.97  8644.547 11388.94
## 2005.3370       9997.129  9098.975 10895.28  8623.521 11370.74
## 2005.3397       9975.876  9076.798 10874.95  8600.855 11350.90
## 2005.3425       9972.995  9072.995 10873.00  8596.564 11349.43
## 2005.3452      10008.798  9107.876 10909.72  8630.957 11386.64
## 2005.3479       9998.552  9096.710 10900.39  8619.303 11377.80
## 2005.3507       9981.076  9078.314 10883.84  8600.421 11361.73
## 2005.3534       9985.707  9082.026 10889.39  8603.646 11367.77
## 2005.3562       9993.544  9088.945 10898.14  8610.079 11377.01
## 2005.3589       9975.344  9069.828 10880.86  8590.477 11360.21
## 2005.3616       9970.452  9064.020 10876.88  8584.184 11356.72
## 2005.3644       9981.491  9074.143 10888.84  8593.823 11369.16
## 2005.3671       9987.355  9079.093 10895.62  8598.289 11376.42
## 2005.3699       9981.449  9072.274 10890.62  8590.986 11371.91
## 2005.3726       9977.285  9067.197 10887.37  8585.426 11369.14
## 2005.3753       9976.672  9065.674 10887.67  8583.420 11369.92
## 2005.3781       9980.080  9068.171 10891.99  8585.435 11374.72
## 2005.3808       9991.751  9078.932 10904.57  8595.715 11387.79
## 2005.3836       9984.588  9070.861 10898.32  8587.163 11382.01
## 2005.3863       9978.438  9063.802 10893.07  8579.624 11377.25
## 2005.3890       9971.923  9056.380 10887.46  8571.722 11372.12
## 2005.3918       9980.646  9064.198 10897.09  8579.059 11382.23
## 2005.3945       9984.189  9066.836 10901.54  8581.218 11387.16
## 2005.3973       9985.440  9067.183 10903.70  8581.086 11389.79
## 2005.4000       9984.413  9065.252 10903.57  8578.678 11390.15
## 2005.4027       9982.578  9062.515 10902.64  8575.462 11389.69
## 2005.4055       9975.932  9054.967 10896.90  8567.437 11384.43
## 2005.4082       9976.285  9054.419 10898.15  8566.413 11386.16
## 2005.4110       9981.684  9058.919 10904.45  8570.436 11392.93
## 2005.4137       9979.056  9055.392 10902.72  8566.434 11391.68
## 2005.4164       9975.768  9051.206 10900.33  8561.772 11389.76
## 2005.4192       9987.543  9062.083 10913.00  8572.174 11402.91
## 2005.4219       9992.549  9066.193 10918.91  8575.810 11409.29
## 2005.4247       9991.107  9063.856 10918.36  8572.999 11409.22
## 2005.4274       9988.917  9060.771 10917.06  8569.440 11408.39
## 2005.4301      10005.768  9076.728 10934.81  8584.924 11426.61
## 2005.4329      10021.061  9091.129 10950.99  8598.852 11443.27
## 2005.4356      10026.901  9096.076 10957.73  8603.327 11450.47
## 2005.4384      10000.499  9068.783 10932.21  8575.562 11425.44
## 2005.4411       9993.378  9060.772 10925.98  8567.080 11419.68
## 2005.4438       9995.309  9061.813 10928.80  8567.650 11422.97
## 2005.4466      10007.782  9073.398 10942.17  8578.765 11436.80
## 2005.4493      10012.734  9077.462 10948.01  8582.359 11443.11
## 2005.4521      10017.521  9081.362 10953.68  8585.789 11449.25
## 2005.4548      10015.338  9078.293 10952.38  8582.251 11448.42
## 2005.4575      10022.710  9084.779 10960.64  8588.269 11457.15
## 2005.4603      10023.532  9084.717 10962.35  8587.739 11459.33
## 2005.4630      10016.269  9076.570 10955.97  8579.124 11453.41
## 2005.4658      10011.425  9070.844 10952.01  8572.930 11449.92
## 2005.4685      10011.596  9070.133 10953.06  8571.753 11451.44
## 2005.4712      10011.038  9068.694 10953.38  8569.847 11452.23
## 2005.4740      10007.009  9063.784 10950.23  8564.471 11449.55
## 2005.4767      10006.301  9062.197 10950.40  8562.418 11450.18
## 2005.4795      10017.119  9072.136 10962.10  8571.893 11462.35
## 2005.4822      10026.781  9080.921 10972.64  8580.213 11473.35
## 2005.4849      10018.937  9072.199 10965.67  8571.027 11466.85
## 2005.4877      10010.590  9062.976 10958.20  8561.340 11459.84
## 2005.4904      10004.486  9055.997 10952.98  8553.897 11455.07
## 2005.4932      10001.377  9052.013 10950.74  8549.451 11453.30
## 2005.4959      10001.496  9051.258 10951.73  8548.233 11454.76
## 2005.4986      10002.880  9051.769 10953.99  8548.282 11457.48
## 2005.5014      10004.290  9052.307 10956.27  8548.358 11460.22
## 2005.5041      10004.582  9051.728 10957.44  8547.317 11461.85
## 2005.5068      10008.735  9055.010 10962.46  8550.139 11467.33
## 2005.5096      10024.934  9070.340 10979.53  8565.008 11484.86
## 2005.5123      10020.065  9064.602 10975.53  8558.810 11481.32
## 2005.5151      10005.839  9049.508 10962.17  8543.256 11468.42
## 2005.5178      10000.523  9043.324 10957.72  8536.614 11464.43
## 2005.5205      10000.422  9042.357 10958.49  8535.187 11465.66
## 2005.5233      10003.214  9044.283 10962.15  8536.655 11469.77
## 2005.5260       9997.079  9037.282 10956.88  8529.197 11464.96
## 2005.5288       9990.041  9029.380 10950.70  8520.837 11459.24
## 2005.5315       9994.288  9032.764 10955.81  8523.764 11464.81
## 2005.5342      10003.981  9041.594 10966.37  8532.137 11475.83
## 2005.5370      10014.152  9050.903 10977.40  8540.990 11487.31
## 2005.5397      10004.604  9040.494 10968.71  8530.125 11479.08
## 2005.5425       9993.205  9028.235 10958.18  8517.410 11469.00
## 2005.5452       9990.594  9024.763 10956.42  8513.484 11467.70
## 2005.5479       9998.378  9031.689 10965.07  8519.955 11476.80
## 2005.5507      10011.893  9044.345 10979.44  8532.157 11491.63
## 2005.5534      10003.111  9034.706 10971.52  8522.064 11484.16
## 2005.5562       9995.095  9025.834 10964.36  8512.738 11477.45
## 2005.5589      10002.943  9032.826 10973.06  8519.277 11486.61
## 2005.5616      10009.816  9038.844 10980.79  8524.842 11494.79
## 2005.5644      10010.432  9038.606 10982.26  8524.152 11496.71
## 2005.5671      10008.449  9035.769 10981.13  8520.863 11496.04
## 2005.5699      10011.653  9038.120 10985.19  8522.763 11500.54
## 2005.5726      10014.235  9039.849 10988.62  8524.041 11504.43
## 2005.5753      10013.402  9038.165 10988.64  8521.906 11504.90
## 2005.5781      10014.407  9038.320 10990.49  8521.610 11507.20
## 2005.5808      10006.270  9029.333 10983.21  8512.174 11500.37
## 2005.5836       9997.285  9019.499 10975.07  8501.890 11492.68
## 2005.5863      10006.399  9027.765 10985.03  8509.707 11503.09
## 2005.5890      10017.764  9038.282 10997.25  8519.775 11515.75
## 2005.5918      10019.154  9038.825 10999.48  8519.870 11518.44
## 2005.5945      10012.502  9031.326 10993.68  8511.923 11513.08
## 2005.5973      10002.450  9020.429 10984.47  8500.578 11504.32
## 2005.6000       9998.981  9016.115 10981.85  8495.818 11502.14
## 2005.6027       9999.372  9015.662 10983.08  8494.917 11503.83
## 2005.6055      10000.718  9016.165 10985.27  8494.974 11506.46
## 2005.6082      10001.000  9015.605 10986.40  8493.968 11508.03
## 2005.6110      10006.234  9019.997 10992.47  8497.914 11514.55
## 2005.6137      10006.487  9019.408 10993.57  8496.880 11516.09
## 2005.6164       9996.248  9008.329 10984.17  8485.357 11507.14
## 2005.6192       9985.690  8996.931 10974.45  8473.514 11497.87
## 2005.6219       9994.671  9005.073 10984.27  8481.212 11508.13
## 2005.6247       9993.909  9003.473 10984.34  8479.168 11508.65
## 2005.6274       9983.652  8992.378 10974.93  8467.630 11499.67
## 2005.6301       9969.655  8977.544 10961.77  8452.353 11486.96
## 2005.6329       9976.776  8983.829 10969.72  8458.195 11495.36
## 2005.6356       9981.599  8987.817 10975.38  8461.741 11501.46
## 2005.6384       9965.202  8970.585 10959.82  8444.067 11486.34
## 2005.6411       9964.504  8969.053 10959.96  8442.094 11486.92
## 2005.6438       9980.673  8984.388 10976.96  8456.987 11504.36
## 2005.6466       9988.298  8991.182 10985.42  8463.340 11513.26
## 2005.6493       9979.885  8981.936 10977.83  8453.654 11506.12
## 2005.6521       9976.530  8977.750 10975.31  8449.028 11504.03
## 2005.6548       9978.180  8978.570 10977.79  8449.408 11506.95
## 2005.6575       9991.070  8990.629 10991.51  8461.028 11521.11
## 2005.6603       9994.148  8992.878 10995.42  8462.837 11525.46
## 2005.6630       9982.251  8980.152 10984.35  8449.674 11514.83
## 2005.6658       9973.850  8970.924 10976.78  8440.007 11507.69
## 2005.6685       9989.636  8985.883 10993.39  8454.528 11524.74
## 2005.6712       9983.209  8978.629 10987.79  8446.837 11519.58
## 2005.6740       9981.434  8976.028 10986.84  8443.798 11519.07
## 2005.6767       9989.096  8982.865 10995.33  8450.198 11527.99
## 2005.6795       9975.739  8968.684 10982.79  8435.581 11515.90
## 2005.6822       9952.052  8944.173 10959.93  8410.634 11493.47
## 2005.6849       9966.710  8958.008 10975.41  8424.034 11509.39
## 2005.6877       9992.065  8982.541 11001.59  8448.131 11536.00
## 2005.6904      10003.955  8993.609 11014.30  8458.764 11549.15
## 2005.6932      10003.516  8992.348 11014.68  8457.069 11549.96
## 2005.6959       9999.879  8987.891 11011.87  8452.177 11547.58
## 2005.6986       9995.315  8982.507 11008.12  8446.359 11544.27
## 2005.7014       9990.310  8976.684 11003.94  8440.102 11540.52
## 2005.7041       9987.340  8972.895 11001.78  8435.880 11538.80
## 2005.7068      10004.192  8988.929 11019.45  8451.482 11556.90
## 2005.7096      10007.390  8991.310 11023.47  8453.429 11561.35
## 2005.7123      10011.434  8994.538 11028.33  8456.225 11566.64
## 2005.7151       9995.684  8977.972 11013.40  8439.228 11552.14
## 2005.7178       9995.321  8976.794 11013.85  8437.618 11553.02
## 2005.7205      10010.256  8990.914 11029.60  8451.307 11569.20
## 2005.7233      10009.830  8989.674 11029.99  8449.636 11570.02
## 2005.7260      10005.418  8984.449 11026.39  8443.981 11566.85
## 2005.7288       9996.196  8974.415 11017.98  8433.517 11558.88
## 2005.7315      10020.831  8998.238 11043.42  8456.909 11584.75
## 2005.7342      10033.839  9010.435 11057.24  8468.677 11599.00
## 2005.7370      10048.582  9024.367 11072.80  8482.180 11614.98
## 2005.7397      10037.252  9012.226 11062.28  8469.611 11604.89
## 2005.7425      10028.806  9002.972 11054.64  8459.928 11597.68
## 2005.7452      10023.244  8996.601 11049.89  8453.129 11593.36
## 2005.7479      10000.145  8972.694 11027.60  8428.794 11571.50
## 2005.7507      10002.818  8974.559 11031.08  8430.232 11575.40
## 2005.7534      10013.159  8984.094 11042.22  8439.339 11586.98
## 2005.7562      10019.619  8989.747 11049.49  8444.566 11594.67
## 2005.7589      10028.618  8997.941 11059.30  8452.334 11604.90
## 2005.7616      10028.287  8996.805 11059.77  8450.771 11605.80
## 2005.7644      10010.075  8977.789 11042.36  8431.330 11588.82
## 2005.7671      10005.387  8972.297 11038.48  8425.412 11585.36
## 2005.7699      10009.607  8975.714 11043.50  8428.404 11590.81
## 2005.7726      10022.912  8988.216 11057.61  8440.481 11605.34
## 2005.7753      10034.756  8999.259 11070.25  8451.100 11618.41
## 2005.7781      10044.769  9008.471 11081.07  8459.888 11629.65
## 2005.7808      10038.720  9001.621 11075.82  8452.614 11624.83
## 2005.7836      10037.199  8999.300 11075.10  8449.870 11624.53
## 2005.7863      10033.306  8994.608 11072.00  8444.755 11621.86
## 2005.7890      10061.768  9022.271 11101.27  8471.995 11651.54
## 2005.7918      10085.104  9044.809 11125.40  8494.110 11676.10
## 2005.7945      10083.046  9041.954 11124.14  8490.833 11675.26
## 2005.7973      10077.970  9036.081 11119.86  8484.538 11671.40
## 2005.8000      10060.887  9018.202 11103.57  8466.238 11655.54
## 2005.8027      10047.291  9003.810 11090.77  8451.425 11643.16
## 2005.8055      10063.800  9019.523 11108.08  8466.717 11660.88
## 2005.8082      10064.381  9019.311 11109.45  8466.084 11662.68
## 2005.8110      10067.327  9021.463 11113.19  8467.816 11666.84
## 2005.8137      10020.251  8973.594 11066.91  8419.527 11620.98
## 2005.8164      10027.278  8979.828 11074.73  8425.341 11629.21
## 2005.8192      10030.949  8982.706 11079.19  8427.800 11634.10
## 2005.8219      10024.192  8975.158 11073.23  8419.833 11628.55
## 2005.8247      10056.264  9006.440 11106.09  8450.696 11661.83
## 2005.8274      10044.581  8993.967 11095.20  8437.805 11651.36
## 2005.8301      10038.023  8986.618 11089.43  8430.038 11646.01
## 2005.8329      10027.330  8975.137 11079.52  8418.139 11636.52
## 2005.8356      10062.992  9010.010 11115.97  8452.595 11673.39
## 2005.8384      10064.429  9010.659 11118.20  8452.827 11676.03
## 2005.8411      10026.766  8972.209 11081.32  8413.961 11639.57
## 2005.8438      10035.869  8980.526 11091.21  8421.860 11649.88
## 2005.8466      10069.399  9013.269 11125.53  8454.188 11684.61
## 2005.8493      10093.408  9036.493 11150.32  8476.996 11709.82
## 2005.8521      10091.462  9033.762 11149.16  8473.849 11709.08
## 2005.8548      10070.571  9012.086 11129.06  8451.758 11689.38
## 2005.8575      10068.548  9009.279 11127.82  8448.536 11688.56
## 2005.8603      10030.662  8970.610 11090.71  8409.453 11651.87
## 2005.8630       9990.480  8929.646 11051.31  8368.074 11612.89
## 2005.8658      10010.923  8949.307 11072.54  8387.321 11634.53
## 2005.8685      10043.698  8981.301 11106.10  8418.901 11668.50
## 2005.8712      10052.981  8989.802 11116.16  8426.989 11678.97
## 2005.8740      10048.018  8984.059 11111.98  8420.833 11675.20
## 2005.8767      10037.934  8973.195 11102.67  8409.557 11666.31
## 2005.8795      10062.520  8997.003 11128.04  8432.952 11692.09
## 2005.8822      10067.274  9000.978 11133.57  8436.515 11698.03
## 2005.8849      10062.958  8995.883 11130.03  8431.008 11694.91
## 2005.8877      10049.891  8982.039 11117.74  8416.752 11683.03
## 2005.8904      10046.852  8978.223 11115.48  8412.525 11681.18
## 2005.8932      10063.268  8993.862 11132.67  8427.754 11698.78
## 2005.8959      10054.851  8984.670 11125.03  8418.151 11691.55
## 2005.8986      10061.361  8990.405 11132.32  8423.475 11699.25
## 2005.9014      10039.769  8968.038 11111.50  8400.698 11678.84
## 2005.9041      10046.557  8974.052 11119.06  8406.303 11686.81
## 2005.9068      10037.986  8964.708 11111.26  8396.548 11679.42
## 2005.9096      10045.275  8971.223 11119.33  8402.655 11687.89
## 2005.9123      10030.917  8956.093 11105.74  8387.116 11674.72
## 2005.9151      10001.777  8926.182 11077.37  8356.796 11646.76
## 2005.9178      10022.099  8945.732 11098.47  8375.938 11668.26
## 2005.9205      10029.009  8951.871 11106.15  8381.669 11676.35
## 2005.9233      10047.419  8969.511 11125.33  8398.901 11695.94
## 2005.9260      10065.511  8986.833 11144.19  8415.816 11715.21
## 2005.9288      10066.752  8987.306 11146.20  8415.881 11717.62
## 2005.9315      10049.229  8969.014 11129.44  8397.182 11701.28
## 2005.9342      10082.606  9001.623 11163.59  8429.385 11735.83
## 2005.9370      10064.194  8982.443 11145.94  8409.799 11718.59
## 2005.9397      10064.438  8981.920 11146.96  8408.870 11720.01
## 2005.9425      10056.260  8972.976 11139.54  8399.520 11713.00
## 2005.9452      10069.058  8985.008 11153.11  8411.146 11726.97
## 2005.9479      10070.393  8985.578 11155.21  8411.311 11729.47
## 2005.9507      10066.078  8980.498 11151.66  8405.826 11726.33
## 2005.9534      10071.786  8985.442 11158.13  8410.366 11733.21
## 2005.9562      10066.063  8978.955 11153.17  8403.474 11728.65
## 2005.9589      10073.946  8986.075 11161.82  8410.190 11737.70
## 2005.9616      10073.379  8984.746 11162.01  8408.458 11738.30
## 2005.9644      10073.981  8984.585 11163.38  8407.894 11740.07
## 2005.9671      10077.150  8986.993 11167.31  8409.898 11744.40
## 2005.9699      10105.396  9014.478 11196.31  8436.981 11773.81
## 2005.9726      10117.362  9025.683 11209.04  8447.784 11786.94
## 2005.9753      10104.318  9011.879 11196.76  8433.577 11775.06
## 2005.9781      10079.751  8986.553 11172.95  8407.849 11751.65
## 2005.9808      10054.534  8960.577 11148.49  8381.471 11727.60
## 2005.9836      10039.199  8944.483 11133.91  8364.976 11713.42
## 2005.9863      10048.477  8953.003 11143.95  8373.095 11723.86
## 2005.9890      10052.697  8956.466 11148.93  8376.157 11729.24
## 2005.9918      10066.834  8969.846 11163.82  8389.136 11744.53
## 2005.9945      10069.695  8971.951 11167.44  8390.841 11748.55
## 2005.9973      10087.467  8988.968 11185.97  8407.457 11767.48
## 2006.0000      10064.366  8965.111 11163.62  8383.201 11745.53
## 2006.0027      10067.201  8967.191 11167.21  8384.882 11749.52
## 2006.0055      10075.013  8974.249 11175.78  8391.540 11758.49

##NOTE: AIC = 32047.14 , BIC = 32064.22 , RMSE = 31.77912
Also, the headline of the plot below contains ETS(A,N,N) showing that the automated model characterized error of the “tsWSR_PK” SERIES as Additative (A), trend as None (A) and seasonality as None (N) which is equivalent to simple exponential model(ses) with only alpha parameter i.e level.

#Model 2 : ARIMA Model

#Changing to stationary set(AutoRegressive Moving Average) It is used for forecasting for both seasonal and non- seasonal level.

#Changing to stationary set

train.sta_Var10 <- diff(train9)
test.sta_Var10 <- diff(test9)

#2.1 Auto Arima model without sesonality:

set.seed(301)
model_Var10 <- auto.arima(train.sta_Var10, seasonal = FALSE)
summary(model_Var10)
## Series: train.sta_Var10 
## ARIMA(3,0,2) with zero mean 
## 
## Coefficients:
##          ar1      ar2      ar3      ma1      ma2
##       0.5810  -0.1244  -0.0149  -0.5463  -0.3725
## s.e.  0.1652   0.1546   0.0731   0.1636   0.1495
## 
## sigma^2 estimated as 979.6:  log likelihood=-10656.84
## AIC=21325.67   AICc=21325.71   BIC=21359.83
## 
## Training set error measures:
##                      ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set -0.1180388 31.26308 22.40424 NaN  Inf 0.6098413 0.001649569

##NOTE: AIC = 21325.67 , BIC = 21359.83 , RMSE = 31.26308

tsdisplay(residuals(model_Var10), lag.max = 45, main = '(1,1,1) Model Residuals')

#2.2 Auto Arima model with sesonality:

set.seed(301)
model2_Var10 <- auto.arima(train.sta_Var10, seasonal= TRUE)
summary(model2_Var10)
## Series: train.sta_Var10 
## ARIMA(3,0,2) with zero mean 
## 
## Coefficients:
##          ar1      ar2      ar3      ma1      ma2
##       0.5810  -0.1244  -0.0149  -0.5463  -0.3725
## s.e.  0.1652   0.1546   0.0731   0.1636   0.1495
## 
## sigma^2 estimated as 979.6:  log likelihood=-10656.84
## AIC=21325.67   AICc=21325.71   BIC=21359.83
## 
## Training set error measures:
##                      ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set -0.1180388 31.26308 22.40424 NaN  Inf 0.6098413 0.001649569

##NOTE: AIC = 21325.67 , BIC = 21359.83 , RMSE = 31.26308

tsdisplay(residuals(model2_Var10), lag.max = 20, main = 'Seasonal Model Residuals')

NOTE: Above Graphs shows serious lags at 13, so modify model for p or q = 13

  1. Manual ARIMA:

Passing parameter order = (3,0,13) which means p(member of lags of a variable to be used as a predictor) = 3, d = 0(members of differences needed for stationarity) and q = 13 (moving average order - number of lagged forecasts error in the prediction equation)

set.seed(301)
model1_Var10 <- arima(train.sta_Var10, order = c(3,0,13))
summary(model1_Var10)
## 
## Call:
## arima(x = train.sta_Var10, order = c(3, 0, 13))
## 
## Coefficients:
##          ar1     ar2      ar3      ma1      ma2     ma3     ma4      ma5
##       0.9497  0.1457  -0.6680  -0.9163  -0.6572  0.8243  0.2687  -0.1887
## s.e.  0.1509  0.2476   0.1478   0.1516   0.2431  0.0766  0.0895   0.0411
##           ma6     ma7      ma8     ma9    ma10     ma11    ma12     ma13
##       -0.1856  -0.031  -0.0403  0.0080  0.0360  -0.0513  0.0410  -0.0296
## s.e.   0.0416   0.041   0.0381  0.0384  0.0402   0.0336  0.0363   0.0248
##       intercept
##          0.0084
## s.e.     0.0910
## 
## sigma^2 estimated as 966.6:  log likelihood = -10644.79,  aic = 21325.57
## 
## Training set error measures:
##                      ME    RMSE      MAE MPE MAPE      MASE         ACF1
## Training set -0.1717054 31.0898 22.29635 NaN  Inf 0.6759088 0.0009877062

##NOTE: AIC = 21325.57 , RMSE = 31.0898

tsdisplay(residuals(model1_Var10), lag.max = 20, main = 'Seasonal Model Residuals')

Note: It seems good fit.

Fit the observed with the predicted values for both models in the same plot.

#Plotting forecast for each model

par(mfrow = c(2,2))
fit_auto_train_Var10 %>% forecast(h=341) %>% autoplot() 

model1_Var10 %>% forecast(h=341) %>% autoplot() 

model2_Var10 %>% forecast(h=341) %>% autoplot() 

model_Var10 %>% forecast(h=341) %>% autoplot() 

COMMENT:
1. The plots includes 80% to 95% confidence interval.
2. For ARIMA models it plots the forecast but because of the nature of autoregressive process of 3 lags i.e. AR3, these forecast stabilizes quite smoothly.

Plot and comment on the residuals of the fitted data for both models in the same plot.

Residual Plot - to confirm no problem with this model

par(mfrow= c(2,2))
hist(fit_auto_train_Var10$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(fit_auto_train_Var10$residuals))

hist(model_Var10$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model_Var10$residuals))

hist(model1_Var10$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model1_Var10$residuals))

hist(model2_Var10$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model2_Var10$residuals))

OBSERVATION: The residuals for each model are normally distributed, which shows the good fit of the models. It is normally distrbuted.

Testing the autocorrelation and partial autocorrelation of the residuals (using the plots and Ljung-Box Q statistic) up to an appropriate lag *

  1. ACF & PACF PLOTS
par(mfrow = c(4,2))
acf(fit_auto_train_Var10$residuals, main = 'Correlogram')
pacf(fit_auto_train_Var10$residuals, main = 'Partial Corelogram')

acf(model_Var10$residuals, main = 'Correlogram')
pacf(model_Var10$residuals, main = 'Partial Corelogram')

acf(model1_Var10$residuals, main = 'Correlogram')
pacf(model1_Var10$residuals, main = 'Partial Corelogram')

acf(model2_Var10$residuals, main = 'Correlogram')
pacf(model2_Var10$residuals, main = 'Partial Corelogram')

NOTE:
1. Residuals are the difference between actaual and fitted values
2. These blue lines are signficance bounds
3. This graph shows us the autocorelations for insample forecast errors
4. Do not exceed these significance bounds for lags 1 to end

  1. Ljunx-Box Test to check the autocorelations of the residuals

HO: No serial correlation upto 20 lags (does not exhibit lack of fit of the model)
Ha: Serial corelation is present (the model exhibits lack of fit)

We reject the null hypothesis, if p value is less than 0.05. This test is generally done for ARIMA models

Box.test(model_Var10$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model_Var10$residuals
## X-squared = 19.094, df = 20, p-value = 0.5157
Box.test(model1_Var10$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model1_Var10$residuals
## X-squared = 6.7111, df = 20, p-value = 0.9975
Box.test(model2_Var10$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model2_Var10$residuals
## X-squared = 19.094, df = 20, p-value = 0.5157

OBSERVATIONS:
For above performed Ljunx-Box Test on ARIMA models, we can say that p-value is more than significance level 0.05, then there is not a statistical significance.
We conclude that there is a liitle evidence of non-zero auto-corelations in the insample forecast errors at lags 1 to 20.
So we fail to rejects the null hypothesis. Hence, we conclude that there is no serial auto corelation present between lags. Hence, the model seems to be good fit.

Test and comment on the normality of the residuals.

Here, further testing the normality by plotting Normal Q-Q plots. From below plots we observed that most of the datapoints lies on the line. Hence, we could conclude that our model produces residuals that are normal

par(mfrow = c(2,2))
qqnorm(fit_auto_train_Var10$residuals)
qqnorm(model_Var10$residuals)
qqnorm(model1_Var10$residuals)
qqnorm(model2_Var10$residuals)

Validating The Model by testing the model performance with Holdout set

Here, choosing RMSE as evaluation parameter. The lower RMSE indicates a more accurate forecast.

Below, is the comparison of model. We are fitting the model to test set which is the holdout set. And checking the accuracy using RMSE(Root mean square error) evaluation parameter.

accuracy(forecast(fit_auto_train_Var10), test9) ["Test set", "RMSE"]
## [1] 157.3138

#o/p - 157.3138

accuracy(forecast(model1_Var10), test.sta_Var10) ["Test set", "RMSE"]
## [1] 35.2608

#O/P - 35.2608

accuracy(forecast(model2_Var10), test.sta_Var10) ["Test set", "RMSE"]
## [1] 35.00271

#o/p - 35.00271

accuracy(forecast(model_Var10), test.sta_Var10) ["Test set", "RMSE"]
## [1] 35.00271
#O/P - 35.00271

NOTE:- Here, the RMSE value is less of ARIMA models compared to exponential model.
The minimum RMSE is of model2_Var10

COMMENT ON BEST MODEL AND ITS PARAMETERS

We observed the best model to be ARIMA model i.e.“model2_Var10”.

summary(model2_Var10)
## Series: train.sta_Var10 
## ARIMA(3,0,2) with zero mean 
## 
## Coefficients:
##          ar1      ar2      ar3      ma1      ma2
##       0.5810  -0.1244  -0.0149  -0.5463  -0.3725
## s.e.  0.1652   0.1546   0.0731   0.1636   0.1495
## 
## sigma^2 estimated as 979.6:  log likelihood=-10656.84
## AIC=21325.67   AICc=21325.71   BIC=21359.83
## 
## Training set error measures:
##                      ME     RMSE      MAE MPE MAPE      MASE        ACF1
## Training set -0.1180388 31.26308 22.40424 NaN  Inf 0.6098413 0.001649569

The best forecasted model is ARIMA(3,0,3) ,which means

p = 3
d = 0
q = 3

For the best model, 3 member of lag are used as a predictor. As we transformed the series to stationary so members of differences needed for stationarity is 0. 3 are the number of lagged forecasts errors used in the prediction equation.

AIC = 21325.67, RMSE = 31.26308 which is minimum as compared to other models.

11. Forecasting on SLP change from previous day(SLP_)

#Checking the attributes of time series

attributes(tsSLP_)
## $tsp
## [1] 1998.00 2004.94  365.00
## 
## $class
## [1] "ts"

#Plotting the series

plot(tsSLP_, main = "SLP change from previous day ", col = "red")

From the plot, we observes that there is absence of trend and seems to have seasonlity.

Checking the statistics and visualising the missing data in the series:

#Visualising the distribution of missing data

plotNA.distribution(tsSLP_)

As we see there is the missing data which is missed at random, so locf and NOCB seems the good technique to implement here.

#Impute the missing values with “na_locf” function of “imputeTS package” and visualise the imputed values in the time series

tsSLP_.imp <- na_locf(tsSLP_)
plotNA.imputations(tsSLP_, tsSLP_.imp)

#Decomposition Of Time Series

decomp10 <- decompose(tsSLP_.imp)
plot(decomp10, col = "red")

OBSERVATIONS:
1)There does not seems to have trend.
2)We observe some seasonality in the series. Seasonality describes cyclical effects due to the time of the year.

Let’s explore this seasonality pattern by plotting the decomp$seasonal components such that y-axis represents Seasonality Index and x-axis has month

#Plotting the decomposed series

plot(decomp10$seasonal[1:365], type ='b', xlab = 'Years', ylab = 'Seasonality Index', col = 'blue', las = 2, main = "SEASONAL DATA")

OBSERVATIONS:

  1. The plot shows a strong and significant sea level presuure in winter and summer.
  2. Also, no increase in the transition seasons.

#Stationarity check of time series We could check the stationarity of time series by ADF and KPSS Tests:

  1. Augmented Dickey- Fuller(ADF) t-statistic test for unit root

If we fail to reject the null hypothesis, we can say that the series is non-stationary. This means that the series can be linear or difference stationary[5]

adf.test(tsSLP_.imp)
## Warning in adf.test(tsSLP_.imp): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  tsSLP_.imp
## Dickey-Fuller = -17.313, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary

NOTE: p-value is smaller, hence stationary

  1. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) for level or trend stationarity

KPSS is another test for checking the stationarity of a time series.The null and alternate hypothesis for the KPSS test are opposite that of the ADF test,

kpss.test(tsSLP_.imp, null="Trend")
## Warning in kpss.test(tsSLP_.imp, null = "Trend"): p-value greater than printed
## p-value
## 
##  KPSS Test for Trend Stationarity
## 
## data:  tsSLP_.imp
## KPSS Trend = 0.0077208, Truncation lag parameter = 8, p-value = 0.1

NOTE: p-value is greater, hence stationary

Plotting ACF to check the auto-corelations

acf(tsSLP_.imp,lag.max = length(tsSLP_.imp),xlab = "lag #", ylab = 'ACF',main='Statiionary Check ')

NOTE: Here, we observe the data lies inside the blue-dotted lines means the significance level.

#CREATING MODELS AND FORECASTING: USING EXPONENTIAL SMOOTHING AND ARIMA TECHNIQUES FOR TIME SERIES FORECASTING

  1. TRAIN- TEST SET SPLIT
train10 <- window(tsSLP_.imp, end = c(2004, 3))
test10 <- window(tsSLP_.imp, start = c(2004, 4))

#Model 1 : Auto Exponential Smoothening Model

set.seed(301)
fit_auto_train_Var11 <- forecast(train10)
summary(fit_auto_train_Var11)
## 
## Forecast method: STL +  ETS(A,N,N)
## 
## Model Information:
## ETS(A,N,N) 
## 
## Call:
##  ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend) 
## 
##   Smoothing parameters:
##     alpha = 1e-04 
## 
##   Initial states:
##     l = -0.2717 
## 
##   sigma:  31.3596
## 
##      AIC     AICc      BIC 
## 31986.86 31986.87 32003.94 
## 
## Error measures:
##                        ME     RMSE      MAE MPE MAPE     MASE     ACF1
## Training set -0.008456595 31.34532 22.85666 NaN  Inf 0.624497 0.197499
## 
## Forecasts:
##           Point Forecast      Lo 80       Hi 80      Lo 95    Hi 95
## 2004.0082  -14.944234224 -55.133205 25.24473648  -76.40796 46.51949
## 2004.0110  -20.043176814 -60.232148 20.14579410  -81.50690 41.42055
## 2004.0137   -6.809591320 -46.998562 33.37937979  -68.27332 54.65414
## 2004.0164  -32.354517174 -72.543488  7.83445414  -93.81824 29.10921
## 2004.0192   24.633695925 -15.555276 64.82266744  -36.83003 86.09742
## 2004.0219   23.466577311 -16.722394 63.65554903  -37.99715 84.93031
## 2004.0247   14.788979412 -25.399993 54.97795133  -46.67475 76.25271
## 2004.0274  -28.466481825 -68.655454 11.72249029  -89.93021 32.99725
## 2004.0301  -17.606309820 -57.795282 22.58266250  -79.07004 43.85742
## 2004.0329   14.978494988 -25.210478 55.16746751  -46.48523 76.44222
## 2004.0356   22.819189972 -17.369783 63.00816269  -38.64454 84.28292
## 2004.0384   -6.800083677 -46.989057 33.38888925  -68.26381 54.66365
## 2004.0411  -25.872324109 -66.061297 14.31664902  -87.33605 35.59141
## 2004.0438  -19.989243985 -60.178217 20.19972934  -81.45297 41.47449
## 2004.0466   15.220560738 -24.968413 55.40953426  -46.24317 76.68429
## 2004.0493    2.149633508 -38.039340 42.33860724  -59.31410 63.61336
## 2004.0521   28.905047653 -11.283926 69.09402158  -32.55868 90.36878
## 2004.0548   37.443407807  -2.745566 77.63238194  -24.02032 98.90714
## 2004.0575  -25.792471566 -65.981446 14.39650277  -87.25620 35.67126
## 2004.0603   12.136299392 -28.052675 52.32527393  -49.32743 73.60003
## 2004.0630   -6.503576024 -46.692551 33.68539871  -67.96731 54.96016
## 2004.0658  -42.344881444 -82.533856 -2.15590651 -103.80861 19.11885
## 2004.0685    4.339557856 -35.849417 44.52853299  -57.12418 65.80329
## 2004.0712   28.676059935 -11.512915 68.86503527  -32.78767 90.13979
## 2004.0740  -17.175299000 -57.364275 23.01367654  -78.63903 44.28844
## 2004.0767   -1.449675262 -41.638651 38.73930048  -62.91341 60.01406
## 2004.0795   22.022511257 -18.166465 62.21148720  -39.44122 83.48625
## 2004.0822  -23.024594976 -63.213571 17.16438117  -84.48833 38.43914
## 2004.0849  -43.723487120 -83.912463 -3.53451078 -105.18722 17.74025
## 2004.0877   24.649349995 -15.539627 64.83832654  -36.81439 86.11309
## 2004.0904    4.281529988 -35.907447 44.47050674  -57.18221 65.74527
## 2004.0932    6.693862215 -33.495115 46.88283916  -54.76987 68.15760
## 2004.0959   14.424905782 -25.764071 54.61388293  -47.03883 75.88864
## 2004.0986    2.955701661 -37.233276 43.14467901  -58.50804 64.41944
## 2004.1014    1.804911884 -38.384066 41.99388944  -59.65883 63.26865
## 2004.1041  -17.729592226 -57.918570 22.45938553  -79.19333 43.73415
## 2004.1068   -3.425587699 -43.614566 36.76339026  -64.88933 58.03815
## 2004.1096  -12.287705389 -52.476684 27.90127277  -73.75144 49.17603
## 2004.1123  -13.144025430 -53.333004 27.04495293  -74.60776 48.31971
## 2004.1151  -30.231154317 -70.420133  9.95782424  -91.69489 31.23258
## 2004.1178  -10.620067064 -50.809046 29.56891170  -72.08381 50.84367
## 2004.1205   29.807457521 -10.381521 69.99643648  -31.65628 91.27120
## 2004.1233   27.930378302 -12.258601 68.11935746  -33.53336 89.39412
## 2004.1260  -15.946130865 -56.135110 24.24284850  -77.40987 45.51761
## 2004.1288   -8.230754177 -48.419734 31.95822539  -69.69449 53.23299
## 2004.1315  -30.915365685 -71.104345  9.27361408  -92.37911 30.54837
## 2004.1342    2.299493601 -37.889486 42.48847357  -59.16425 63.76323
## 2004.1370   31.333387793  -8.855592 71.52236796  -30.13035 92.79713
## 2004.1397   24.797570069 -15.391410 64.98655044  -36.66617 86.26131
## 2004.1425  -19.251055970 -59.440037 20.93792460  -80.71480 42.21269
## 2004.1452  -12.401040702 -52.590021 27.78794007  -73.86478 49.06270
## 2004.1479  -19.955345295 -60.144326 20.23363568  -81.41909 41.50840
## 2004.1507  -41.661244932 -81.850226 -1.47226376 -103.12499 19.80250
## 2004.1534   12.771660600 -27.417321 52.96064198  -48.69208 74.23540
## 2004.1562    2.346785367 -37.842196 42.53576695  -59.11696 63.81053
## 2004.1589  -10.630875827 -50.819858 29.55810595  -72.09462 50.83287
## 2004.1616   20.761873796 -19.427108 60.95085578  -40.70187 82.22562
## 2004.1644   12.693208407 -27.495774 52.88219059  -48.77054 74.15695
## 2004.1671  -39.783954451 -79.972937  0.40502793 -101.24770 21.67979
## 2004.1699    0.125219691 -40.063763 40.31420228  -61.33853 61.58896
## 2004.1726   19.483471594 -20.705511 59.67245438  -41.98027 80.94722
## 2004.1753    4.036089848 -36.152893 44.22507284  -57.42766 65.49984
## 2004.1781  -30.292465859 -70.481449  9.89651733  -91.75621 31.17128
## 2004.1808    2.078989234 -38.109994 42.26797262  -59.38476 63.54274
## 2004.1836   34.672236691  -5.516747 74.86122028  -26.79151 96.13598
## 2004.1863   28.730869540 -11.458114 68.91985333  -32.73288 90.19462
## 2004.1890    1.164042646 -39.024941 41.35302664  -60.29970 62.62779
## 2004.1918  -26.004463469 -66.193448 14.18452073  -87.46821 35.45928
## 2004.1945   -7.378041996 -47.567026 32.81094240  -68.84179 54.08571
## 2004.1973   -2.900796230 -43.089781 37.28818837  -64.36454 58.56295
## 2004.2000   -2.858526386 -43.047511 37.33045841  -64.32227 58.60522
## 2004.2027  -32.512216400 -72.701201  7.67676860  -93.97596 28.95153
## 2004.2055  -45.220340630 -85.409326 -5.03135543 -106.68409 16.24341
## 2004.2082    7.382217533 -32.806768 47.57120294  -54.08153 68.84597
## 2004.2110   26.843974367 -13.345011 67.03295997  -34.61978 88.30772
## 2004.2137   18.662595375 -21.526390 58.85158118  -42.80115 80.12635
## 2004.2164  -17.725392548 -57.914379 22.46359346  -79.18914 43.73836
## 2004.2192    1.646948641 -38.542038 41.83593485  -59.81680 63.11070
## 2004.2219   37.725771774  -2.463215 77.91475818  -23.73798 99.18952
## 2004.2247   21.701605209 -18.487381 61.89059182  -39.76215 83.16536
## 2004.2274   -0.046163600 -40.235150 40.14282321  -61.50991 61.41759
## 2004.2301  -15.712653958 -55.901641 24.47633306  -77.17641 45.75110
## 2004.2329  -11.008156055 -51.197143 29.18083116  -72.47191 50.45560
## 2004.2356   13.548278587 -26.640709 53.73726600  -47.91547 75.01203
## 2004.2384    4.745708874 -35.443279 44.93469649  -56.71804 66.20946
## 2004.2411  -19.765347525 -59.954335 20.42364029  -81.22910 41.69841
## 2004.2438    4.037778863 -36.151209 44.22676688  -57.42597 65.50153
## 2004.2466    5.622008468 -34.566980 45.81099669  -55.84175 67.08576
## 2004.2493   -2.019117282 -42.208106 38.16987114  -63.48287 59.44464
## 2004.2521  -30.234103589 -70.423092  9.95488503  -91.69786 31.22965
## 2004.2548  -11.838404799 -52.027394 28.35058403  -73.30216 49.62535
## 2004.2575   20.566094177 -19.622895 60.75508320  -40.89766 82.02985
## 2004.2603   29.835355832 -10.353633 70.02434506  -31.62840 91.29911
## 2004.2630   -2.148655290 -42.337645 38.04033414  -63.61241 59.31510
## 2004.2658  -11.185791576 -51.374781 29.00319805  -72.64955 50.27796
## 2004.2685  -19.836195809 -60.025186 20.35279402  -81.29995 41.62756
## 2004.2712    0.418781954 -39.770208 40.60777199  -61.04497 61.88254
## 2004.2740   11.666211928 -28.522778 51.85520216  -49.79754 73.12997
## 2004.2767   -0.073430186 -40.262421 40.11556025  -61.53719 61.39033
## 2004.2795   36.644505799  -3.544485 76.83349644  -24.81925 98.10826
## 2004.2822   -3.517731394 -43.706722 36.67125944  -64.98149 57.94603
## 2004.2849  -14.133310024 -54.322301 26.05568101  -75.59707 47.33045
## 2004.2877   -3.573322897 -43.762314 36.61566834  -65.03708 57.89044
## 2004.2904   -0.515860914 -40.704852 39.67313053  -61.97962 60.94790
## 2004.2932    5.329225908 -34.859766 45.51821755  -56.13453 66.79298
## 2004.2959  -10.149432995 -50.338425 30.03955885  -71.61319 51.31433
## 2004.2986  -13.823662748 -54.012655 26.36532930  -75.28742 47.64010
## 2004.3014    8.446314707 -31.742678 48.63530695  -53.01744 69.91007
## 2004.3041   13.844330236 -26.344662 54.03332268  -47.61943 75.30809
## 2004.3068    3.791095595 -36.397897 43.98008824  -57.67266 65.25486
## 2004.3096  -12.459446645 -52.648439 27.72954621  -73.92321 49.00431
## 2004.3123    5.684866444 -34.504127 45.87385950  -55.77889 67.14863
## 2004.3151    1.027257069 -39.161736 41.21625032  -60.43650 62.49102
## 2004.3178  -10.470796068 -50.659790 29.71819739  -71.93456 50.99297
## 2004.3205    2.538472225 -37.650521 42.72746588  -58.92529 64.00223
## 2004.3233    1.904192248 -38.284802 42.09318611  -59.55957 63.36795
## 2004.3260   -5.038257764 -45.227252 35.15073629  -66.50202 56.42550
## 2004.3288    8.294144969 -31.894849 48.48313923  -53.16962 69.75791
## 2004.3315   23.526178635 -16.662816 63.71517310  -37.93758 84.98994
## 2004.3342    2.893860553 -37.295134 43.08285522  -58.56990 64.35762
## 2004.3370  -14.869379033 -55.058374 25.31961583  -76.33314 46.59438
## 2004.3397  -21.294848694 -61.483844 18.89414637  -82.75861 40.16892
## 2004.3425   -2.922297359 -43.111293 37.26669791  -64.38606 58.54147
## 2004.3452   19.434507460 -20.754488 59.62350293  -42.02926 80.89827
## 2004.3479  -10.286921224 -50.475917 29.90207444  -71.75069 51.17684
## 2004.3507  -17.516077712 -57.705074 22.67291816  -78.97984 43.94769
## 2004.3534    4.590598236 -35.598398 44.77959431  -56.87317 66.05436
## 2004.3562    6.062425054 -34.126571 46.25142133  -55.40134 67.52619
## 2004.3589  -11.299568257 -51.488565 28.88942822  -72.76333 50.16420
## 2004.3616   -4.931292892 -45.120290 35.25770378  -66.39506 56.53247
## 2004.3644   10.999965659 -29.189031 51.18896254  -50.46380 72.46373
## 2004.3671    8.745706509 -31.443291 48.93470359  -52.71806 70.20947
## 2004.3699    4.763507401 -35.425490 44.95250468  -56.70026 66.22727
## 2004.3726   -4.202190244 -44.391188 35.98680724  -65.66596 57.26158
## 2004.3753   -0.649345017 -40.838343 39.53965266  -62.11311 60.81442
## 2004.3781    3.370809294 -36.818189 43.55980718  -58.09296 64.83458
## 2004.3808   11.633735944 -28.555262 51.82273403  -49.83003 73.09750
## 2004.3836   -7.199646116 -47.388644 32.98935217  -68.66341 54.26412
## 2004.3863   -6.187807773 -46.376806 34.00119071  -67.65158 55.27596
## 2004.3890   -6.551566742 -46.740565 33.63743195  -68.01534 54.91220
## 2004.3918    8.686917821 -31.502081 48.87591671  -52.77685 70.15069
## 2004.3945    3.947583503 -36.241416 44.13658259  -57.51619 65.41135
## 2004.3973    1.214615833 -38.974383 41.40361513  -60.24915 62.67839
## 2004.4000   -1.062335916 -41.251335 39.12666358  -62.52611 60.40143
## 2004.4027   -1.870528889 -42.059529 38.31847081  -63.33430 59.59324
## 2004.4055   -5.313753388 -45.502753 34.87524651  -66.77752 56.15002
## 2004.4082    1.686078435 -38.502922 41.87507853  -59.77769 63.14985
## 2004.4110    6.732946270 -33.456054 46.92194657  -54.73083 68.19672
## 2004.4137   -4.712038561 -44.901039 35.47696194  -66.17581 56.75173
## 2004.4164   -3.321199016 -43.510200 36.86780169  -64.78497 58.14257
## 2004.4192   11.739433482 -28.449567 51.92843438  -49.72434 73.20321
## 2004.4219    2.369557402 -37.819444 42.55855851  -59.09422 63.83333
## 2004.4247   -2.344310137 -42.533311 37.84469117  -63.80808 59.11946
## 2004.4274   -4.828056199 -45.017058 35.36094531  -66.29183 56.63572
## 2004.4301    3.804976526 -36.384025 43.99397823  -57.65880 65.26875
## 2004.4329   15.258119311 -24.930883 55.44712122  -46.20566 76.72189
## 2004.4356    4.041047614 -36.147954 44.23004972  -57.42273 65.50482
## 2004.4384  -20.267566373 -60.456569 19.92143594  -81.73134 41.19621
## 2004.4411   -7.155687836 -47.344690 33.03331468  -68.61946 54.30809
## 2004.4438    1.895808570 -38.293194 42.08481128  -59.56797 63.35958
## 2004.4466   12.439120205 -27.749883 52.62812312  -49.02466 73.90290
## 2004.4493    4.917837084 -35.271166 45.10684020  -56.54594 66.38161
## 2004.4521    4.752437341 -35.436566 44.94144066  -56.71134 66.21621
## 2004.4548   -0.918585749 -41.107589 39.27041777  -62.38236 60.54519
## 2004.4575    0.890125652 -39.298878 41.07912937  -60.57365 62.35390
## 2004.4603    0.788916716 -39.400087 40.97792064  -60.67486 62.25269
## 2004.4630   -2.224609855 -42.413614 37.96439427  -63.68839 59.23917
## 2004.4658    0.195491553 -39.993513 40.38449588  -61.26829 61.65927
## 2004.4685   10.283187554 -29.905817 50.47219208  -51.18059 71.74697
## 2004.4712   -0.590986679 -40.779991 39.59801805  -62.05477 60.87279
## 2004.4740   -4.062571494 -44.251576 36.12643343  -65.52635 57.40121
## 2004.4767   -0.740515476 -40.929521 39.44848965  -62.20429 60.72326
## 2004.4795   10.785741737 -29.403264 50.97474707  -50.67804 72.24952
## 2004.4822    6.895598254 -33.293407 47.08460379  -54.56818 68.35938
## 2004.4849   -7.876892225 -48.065898 32.31211351  -69.34067 53.58689
## 2004.4877   -8.378733784 -48.567740 31.81027215  -69.84251 53.08505
## 2004.4904   -6.135640188 -46.324646 34.05336595  -67.59942 55.32814
## 2004.4932   -3.140599854 -43.329606 37.04840648  -64.60438 58.32318
## 2004.4959    4.872664912 -35.316342 45.06167145  -56.59112 66.33645
## 2004.4986    0.471194505 -39.717812 40.66020125  -60.99259 61.93498
## 2004.5014    1.819876977 -38.369130 42.00888392  -59.64391 63.28366
## 2004.5041    0.260866590 -39.928141 40.44987373  -61.20292 61.72465
## 2004.5068    0.652872013 -39.536135 40.84187936  -60.81091 62.11665
## 2004.5096    3.158157520 -37.030850 43.34716506  -58.30563 64.62194
## 2004.5123   -4.899607996 -45.088616 35.28939975  -66.36339 56.56418
## 2004.5151  -14.255848780 -54.444857 25.93315917  -75.71963 47.20793
## 2004.5178   -5.346161444 -45.535170 34.84284671  -66.80995 56.11762
## 2004.5205   -0.130601655 -40.319610 40.05840670  -61.59439 61.33318
## 2004.5233   -1.429946630 -41.618955 38.75906192  -62.89373 60.03384
## 2004.5260   -3.520436698 -43.709445 36.66857206  -64.98422 57.94335
## 2004.5288   -4.423084011 -44.612093 35.76592494  -65.88687 57.04070
## 2004.5315    6.860193016 -33.328816 47.04920217  -54.60359 68.32398
## 2004.5342   12.305945063 -27.883064 52.49495442  -49.15784 73.76973
## 2004.5370    8.377967953 -31.811042 48.56697751  -53.08582 69.84175
## 2004.5397   -9.578199522 -49.767209 30.61081024  -71.04199 51.88559
## 2004.5425  -11.869938832 -52.058949 28.31907113  -73.33373 49.59385
## 2004.5452   -6.124843841 -46.313854 34.06416632  -67.58863 55.33894
## 2004.5479    5.153006412 -35.036004 45.34201678  -56.31078 66.61679
## 2004.5507    5.680054223 -34.508956 45.86906479  -55.78373 67.14384
## 2004.5534   -8.809371753 -48.998383 31.37963901  -70.27316 52.65442
## 2004.5562   -8.043412560 -48.232424 32.14559841  -69.50720 53.42038
## 2004.5589    6.938560688 -33.250450 47.12757186  -54.52523 68.40235
## 2004.5616    0.317952259 -39.871059 40.50696363  -61.14584 61.78174
## 2004.5644    0.589263475 -39.599748 40.77827505  -60.87453 62.05305
## 2004.5671    0.040700618 -40.148311 40.22971239  -61.42309 61.50449
## 2004.5699   -1.392503849 -41.581516 38.79650812  -62.85629 60.07129
## 2004.5726  -10.096538067 -50.285550 30.09247411  -71.56033 51.36725
## 2004.5753    1.172272123 -39.016740 41.36128450  -60.29152 62.63606
## 2004.5781    3.011391289 -37.177621 43.20040387  -58.45240 64.47518
## 2004.5808   -5.249462129 -45.438475 34.93955065  -66.71325 56.21433
## 2004.5836   -0.009372358 -40.198385 40.17964062  -61.47316 61.45442
## 2004.5863    9.973797669 -30.215216 50.16281085  -51.48999 71.43759
## 2004.5890   11.337201906 -28.851811 51.52621529  -50.12659 72.80099
## 2004.5918    1.363517866 -38.825496 41.55253145  -60.10027 62.82731
## 2004.5945   -6.679088233 -46.868102 33.50992555  -68.14288 54.78470
## 2004.5973  -10.078155251 -50.267169 30.11085874  -71.54195 51.38564
## 2004.6000   -3.494046882 -43.683061 36.69496731  -64.95784 57.96975
## 2004.6027    0.365457595 -39.823557 40.55447198  -61.09834 61.82925
## 2004.6055    1.321938136 -38.867076 41.51095273  -60.14186 62.78573
## 2004.6082    0.259100699 -39.929914 40.44811549  -61.20469 61.72289
## 2004.6110    5.211858549 -34.977156 45.40087354  -56.25194 66.67565
## 2004.6137    0.230382593 -39.958633 40.41939779  -61.23341 61.69418
## 2004.6164  -11.280004609 -51.469020 28.90901079  -72.74380 50.18379
## 2004.6192   -9.558929989 -49.747946 30.63008561  -71.02273 51.90487
## 2004.6219    8.960930222 -31.228086 49.14994602  -52.50287 70.42473
## 2004.6247   -0.781462540 -40.970479 39.40755346  -62.24526 60.68233
## 2004.6274  -10.276035813 -50.465052 29.91298039  -71.73983 51.18776
## 2004.6301  -14.015221099 -54.204238 26.17379530  -75.47902 47.44858
## 2004.6329    7.102783580 -33.086233 47.29180018  -54.36101 68.56658
## 2004.6356    4.806113672 -35.382903 44.99513048  -56.65768 66.26991
## 2004.6384  -16.413671966 -56.602689 23.77534504  -77.87747 45.05013
## 2004.6411   -0.714025543 -40.903043 39.47499166  -62.17782 60.74977
## 2004.6438   16.152691648 -24.036326 56.34170906  -45.31111 77.61649
## 2004.6466   10.212896722 -29.976121 50.40191433  -51.25090 71.67670
## 2004.6493   -6.692916168 -46.881934 33.49610164  -68.15671 54.77088
## 2004.6521   -3.369177506 -43.558196 36.81984051  -64.83298 58.09462
## 2004.6548    1.636830061 -38.552188 41.82584828  -59.82697 63.10063
## 2004.6575   12.876453072 -27.312565 53.06547149  -48.58735 74.34025
## 2004.6603    3.065502726 -37.123516 43.25452134  -58.39830 64.52930
## 2004.6630  -11.041156105 -51.230175 29.14786271  -72.50496 50.42264
## 2004.6658   -6.676852735 -46.865872 33.51216628  -68.14065 54.78695
## 2004.6685   15.775552886 -24.413466 55.96457211  -45.68825 77.23935
## 2004.6712   -6.437295515 -46.626315 33.75172391  -67.90110 55.02651
## 2004.6740   -4.519579817 -44.708599 35.66943981  -65.98338 56.94422
## 2004.6767   -0.550572890 -40.739593 39.63844693  -62.01437 60.91323
## 2004.6795  -13.364800765 -53.553821 26.82421926  -74.82860 48.09900
## 2004.6822  -23.695929400 -63.884950 16.49309083  -85.15973 37.76787
## 2004.6849   14.651046788 -25.537974 54.84006722  -46.81276 76.11485
## 2004.6877   25.348432274 -14.840588 65.53745290  -36.11537 86.81224
## 2004.6904   11.883488878 -28.305532 52.07250971  -49.58031 73.34729
## 2004.6932   -0.444430854 -40.633452 39.74459018  -61.90823 61.01937
## 2004.6959   -3.640901748 -43.829923 36.54811948  -65.10471 57.82290
## 2004.6986   -7.302176963 -47.491198 32.88684447  -68.76598 54.16163
## 2004.7014   -3.641106308 -43.830128 36.54791533  -65.10491 57.82270
## 2004.7041   -2.972993469 -43.162015 37.21602837  -64.43680 58.49081
## 2004.7068   -6.482059185 -46.671081 33.70696285  -67.94586 54.98175
## 2004.7096    3.197266485 -36.991756 43.38628872  -58.26654 64.66107
## 2004.7123    4.045585404 -36.143437 44.23460785  -57.41822 65.50939
## 2004.7151  -15.747323203 -55.936346 24.44169944  -77.21113 45.71648
## 2004.7178   -0.359501137 -40.548524 39.82952171  -61.82331 61.10431
## 2004.7205   14.939584253 -25.249439 55.12860730  -46.52422 76.40339
## 2004.7233   -0.420247627 -40.609271 39.76877562  -61.88405 61.04356
## 2004.7260   -4.405194451 -44.594218 35.78382900  -65.86900 57.05861
## 2004.7288   -7.162702058 -47.351726 33.02632159  -68.62651 54.30111
## 2004.7315   29.473710313 -10.715314 69.66273416  -31.99010 90.93752
## 2004.7342    7.289043929 -32.899980 47.47806798  -54.17476 68.75285
## 2004.7370   14.753126927 -25.435897 54.94215118  -46.71068 76.21694
## 2004.7397  -11.318599800 -51.507624 28.87042465  -72.78241 50.14521
## 2004.7425   -8.433247773 -48.622272 31.75577688  -69.89706 53.03056
## 2004.7452   -5.549215673 -45.738241 34.63980918  -67.01303 55.91459
## 2004.7479  -23.085363008 -63.274388 17.10366205  -84.54917 38.37845
## 2004.7507    2.687074838 -37.501950 42.87610010  -58.77674 64.15088
## 2004.7534   10.355443743 -29.833582 50.54446920  -51.10837 71.81925
## 2004.7562    6.474768021 -33.714258 46.66379368  -54.98904 67.93858
## 2004.7589    9.017072063 -31.171954 49.20609793  -52.44674 70.48088
## 2004.7616   -0.313049178 -40.502075 39.87597689  -61.77686 61.15076
## 2004.7644  -18.192051078 -58.381077 21.99697519  -79.65586 43.27176
## 2004.7671   -4.668004332 -44.857031 35.52102213  -66.13182 56.79581
## 2004.7699    4.242212012 -35.946815 44.43123868  -57.22160 65.70602
## 2004.7726   13.326952157 -26.862075 53.51597903  -48.13686 74.79076
## 2004.7753   11.868349688 -28.320677 52.05737676  -49.59546 73.33216
## 2004.7781   10.038751220 -30.150276 50.22777849  -51.42506 71.50256
## 2004.7808   -6.022435168 -46.211463 34.16659231  -67.48625 55.44138
## 2004.7836   -1.493373840 -41.682402 38.69565383  -62.95719 59.97044
## 2004.7863   -3.863657776 -44.052686 36.32537010  -65.32747 57.60016
## 2004.7890   28.491806026 -11.697222 68.68083410  -32.97201 89.95562
## 2004.7918   23.366433533 -16.822595 63.55546181  -38.09738 84.83025
## 2004.7945   -2.025033965 -42.214062 38.16399451  -63.48885 59.43878
## 2004.7973  -10.115874669 -50.304903 30.07315401  -71.57969 51.34794
## 2004.8000  -22.120572462 -62.309601 18.06845642  -83.58439 39.34324
## 2004.8027  -18.633097093 -58.822126 21.55593199  -80.09691 42.83072
## 2004.8055   11.472170479 -28.716859 51.66119976  -49.99165 72.93599
## 2004.8082   -4.452967296 -44.641997 35.73606219  -65.91678 57.01085
## 2004.8110   -2.088101534 -42.277131 38.10092815  -63.55192 59.37572
## 2004.8137  -30.804157815 -70.993188  9.38487207  -92.26798 30.65966
## 2004.8164    7.067409895 -33.121620 47.25643998  -54.39641 68.53123
## 2004.8192    3.712546384 -36.476484 43.90157668  -57.75127 65.17636
## 2004.8219    7.572529095 -32.616501 47.76155959  -53.89129 69.03635
## 2004.8247   28.150278060 -12.038753 68.33930875  -33.31354 89.61410
## 2004.8274  -15.603545637 -55.792577 24.58548526  -77.06736 45.86027
## 2004.8301  -16.207408604 -56.396440 23.98162249  -77.67123 45.25641
## 2004.8329  -10.645241814 -50.834273 29.54378948  -72.10906 50.81858
## 2004.8356   35.710018195  -4.479013 75.89904969  -25.75380 97.17384
## 2004.8384    3.689909279 -36.499122 43.87894098  -57.77391 65.15373
## 2004.8411  -35.408441562 -75.597473  4.78059034  -96.87226 26.05538
## 2004.8438   11.358040410 -28.830992 51.54707251  -50.10578 72.82186
## 2004.8466   38.429973769  -1.759059 78.61900607  -23.03385 99.89379
## 2004.8493   24.063058391 -16.125974 64.25209090  -37.40076 85.52688
## 2004.8521   -5.787126206 -45.976159 34.40190650  -67.25095 55.67670
## 2004.8548  -22.784474795 -62.973508 17.40455811  -84.24830 38.67935
## 2004.8575   -1.968019962 -42.157053 38.22101315  -63.43184 59.49580
## 2004.8603  -37.829400588 -78.018434  2.35963272  -99.29322 23.63442
## 2004.8630  -40.123969149 -80.313003  0.06506436 -101.58779 21.33985
## 2004.8658   20.501802312 -19.687231 60.69083603  -40.96202 81.96563
## 2004.8685   35.038623064  -5.150411 75.22765698  -26.42520 96.50245
## 2004.8712   13.309700314 -26.879334 53.49873443  -48.15412 74.77352
## 2004.8740   -4.899793566 -45.088828 35.28924075  -66.36362 56.56403
## 2004.8767  -10.019477766 -50.208512 30.16955675  -71.48330 51.44435
## 2004.8795   24.652504530 -15.536530 64.84153925  -36.81132 86.11633
## 2004.8822    4.821624372 -35.367411 45.01065929  -56.64220 66.28545
## 2004.8849   -4.247618128 -44.436653 35.94141699  -65.71144 57.21621
## 2004.8877  -12.995706907 -53.184742 27.19332842  -74.45953 48.46812
## 2004.8904   -2.966753841 -43.155789 37.22228168  -64.43058 58.49707
## 2004.8932   16.489091830 -23.699944 56.67812756  -44.97473 77.95292
## 2004.8959   -8.341499713 -48.530536 31.84753622  -69.80533 53.12233
## 2004.8986    6.585693797 -33.603342 46.77472993  -54.87813 68.04952
## 2004.9014  -21.514480942 -61.703517 18.67455539  -82.97831 39.94935
## 2004.9041    6.866791027 -33.322246 47.05582756  -54.59704 68.33062
## 2004.9068   -8.491147167 -48.680184 31.69788957  -69.95497 52.97268
## 2004.9096    7.366899570 -32.822137 47.55593650  -54.09693 68.83073
## 2004.9123  -14.278205782 -54.467243 25.91083135  -75.74203 47.18562
## 2004.9151  -29.059196359 -69.248234 11.12984098  -90.52302 32.40463
## 2004.9178   20.403320040 -19.785717 60.59235758  -41.06051 81.86715
## 2004.9205    6.991772363 -33.197265 47.18081010  -54.47206 68.45560
## 2004.9233   18.493458337 -21.695580 58.68249628  -42.97037 79.95729
## 2004.9260   18.176142134 -22.012896 58.36518028  -43.28769 79.63997
## 2004.9288   12.451507815 -27.737531 52.64054616  -49.01232 73.91534
## 2004.9315    1.251708469 -38.937330 41.44074701  -60.21212 62.71554
## 2004.9342   33.475024666  -6.714014 73.66406341  -27.98881 94.93886
## 2004.9370  -18.313315940 -58.502355 21.87572301  -79.77715 43.15052
## 2004.9397   -4.757711441 -44.946751 35.43132771  -66.22154 56.70612
## 2004.9425   -2.387925109 -42.576964 37.80111424  -63.85176 59.07591
## 2004.9452   15.986501530 -24.202538 56.17554108  -45.47733 77.45033
## 2004.9479    1.441643683 -38.747396 41.63068343  -60.02219 62.90548
## 2004.9507   -4.207474292 -44.396514 35.98156566  -65.67131 57.25636
## 2004.9534    5.818011203 -34.371029 46.00705136  -55.64582 67.28184
## 2004.9562   -5.612617543 -45.801658 34.57642281  -67.07645 55.85122
## 2004.9589    7.995900077 -32.193140 48.18494063  -53.46793 69.45973
## 2004.9616    0.870249733 -39.318791 41.05929049  -60.59358 62.33408
## 2004.9644   -1.485733515 -41.674774 38.70330744  -62.94957 59.97810
## 2004.9671    4.522164280 -35.666877 44.71120544  -56.94167 65.98600
## 2004.9699   20.927021402 -19.262020 61.11606276  -40.53681 82.39086
## 2004.9726   10.719934566 -29.469107 50.90897613  -50.74390 72.18377
## 2004.9753  -17.023367657 -57.212409 23.16567411  -78.48720 44.44047
## 2004.9781  -28.848344045 -69.037386 11.34069792  -90.31218 32.61549
## 2004.9808  -32.141652930 -72.330695  8.04738924  -93.60549 29.32218
## 2004.9836  -15.207289173 -55.396332 24.98175320  -76.67113 46.25655
## 2004.9863    2.797376773 -37.391666 42.98641934  -58.66646 64.26121
## 2004.9890    4.351510375 -35.837532 44.54055315  -57.11233 65.81535
## 2004.9918   14.270432014 -25.918611 54.45947499  -47.19341 75.73427
## 2004.9945   -2.076693282 -42.265736 38.11234989  -63.54053 59.38714
## 2004.9973   12.836645857 -27.352398 53.02568923  -48.62719 74.30048
## 2005.0000  -16.715050963 -56.904095 23.47399261  -78.17889 44.74879
## 2005.0027    3.020884290 -37.168159 43.20992807  -58.44295 64.48472
## 2005.0055    7.999234299 -32.189810 48.18827828  -53.46460 69.46307
## 2005.0082  -14.944234224 -55.133278 25.24480996  -76.40807 46.51960
## 2005.0110  -20.043176814 -60.232221 20.14586757  -81.50702 41.42066
## 2005.0137   -6.809591320 -46.998636 33.37945326  -68.27343 54.65425
## 2005.0164  -32.354517174 -72.543562  7.83452761  -93.81836 29.10932
## 2005.0192   24.633695925 -15.555349 64.82274091  -36.83014 86.09754
## 2005.0219   23.466577311 -16.722468 63.65562250  -37.99726 84.93042
## 2005.0247   14.788979412 -25.400066 54.97802480  -46.67486 76.25282
## 2005.0274  -28.466481825 -68.655527 11.72256376  -89.93032 32.99736
## 2005.0301  -17.606309820 -57.795356 22.58273597  -79.07015 43.85753
## 2005.0329   14.978494988 -25.210551 55.16754098  -46.48535 76.44234
## 2005.0356   22.819189972 -17.369856 63.00823617  -38.64465 84.28303
## 2005.0384   -6.800083677 -46.989130 33.38896272  -68.26393 54.66376
## 2005.0411  -25.872324109 -66.061371 14.31672249  -87.33617 35.59152
## 2005.0438  -19.989243985 -60.178291 20.19980281  -81.45309 41.47460
## 2005.0466   15.220560738 -24.968486 55.40960774  -46.24328 76.68440
## 2005.0493    2.149633508 -38.039414 42.33868071  -59.31421 63.61348
## 2005.0521   28.905047653 -11.284000 69.09409505  -32.55880 90.36889
## 2005.0548   37.443407807  -2.745640 77.63245541  -24.02044 98.90725
## 2005.0575  -25.792471566 -65.981519 14.39657624  -87.25632 35.67137
## 2005.0603   12.136299392 -28.052749 52.32534740  -49.32755 73.60014
## 2005.0630   -6.503576024 -46.692624 33.68547218  -67.96742 54.96027
## 2005.0658  -42.344881444 -82.533930 -2.15583304 -103.80873 19.11896
## 2005.0685    4.339557856 -35.849491 44.52860647  -57.12429 65.80340
## 2005.0712   28.676059935 -11.512989 68.86510875  -32.78779 90.13991
## 2005.0740  -17.175299000 -57.364348 23.01375001  -78.63915 44.28855
## 2005.0767   -1.449675262 -41.638724 38.73937395  -62.91352 60.01417
## 2005.0795   22.022511257 -18.166538 62.21156067  -39.44134 83.48636
## 2005.0822  -23.024594976 -63.213645 17.16445464  -84.48844 38.43925
## 2005.0849  -43.723487120 -83.912537 -3.53443730 -105.18733 17.74036
## 2005.0877   24.649349995 -15.539700 64.83840001  -36.81450 86.11320
## 2005.0904    4.281529988 -35.907520 44.47058021  -57.18232 65.74538
## 2005.0932    6.693862215 -33.495188 46.88291264  -54.76999 68.15771
## 2005.0959   14.424905782 -25.764145 54.61395640  -47.03894 75.88875
## 2005.0986    2.955701661 -37.233349 43.14475248  -58.50815 64.41955
## 2005.1014    1.804911884 -38.384139 41.99396291  -59.65894 63.26876
## 2005.1041  -17.729592226 -57.918643 22.45945900  -79.19344 43.73426
## 2005.1068   -3.425587699 -43.614639 36.76346373  -64.88944 58.03826
## 2005.1096  -12.287705389 -52.476757 27.90134624  -73.75156 49.17615
## 2005.1123  -13.144025430 -53.333077 27.04502640  -74.60788 48.31983
## 2005.1151  -30.231154317 -70.420206  9.95789771  -91.69501 31.23270
## 2005.1178  -10.620067064 -50.809119 29.56898517  -72.08392 50.84378
## 2005.1205   29.807457521 -10.381595 69.99650995  -31.65639 91.27131
## 2005.1233   27.930378302 -12.258674 68.11943094  -33.53347 89.39423
## 2005.1260  -15.946130865 -56.135184 24.24292197  -77.40998 45.51772
## 2005.1288   -8.230754177 -48.419807 31.95829886  -69.69461 53.23310
## 2005.1315  -30.915365685 -71.104419  9.27368755  -92.37922 30.54849
## 2005.1342    2.299493601 -37.889560 42.48854704  -59.16436 63.76335
## 2005.1370   31.333387793  -8.855666 71.52244143  -30.13047 92.79724
## 2005.1397   24.797570069 -15.391484 64.98662391  -36.66628 86.26142
## 2005.1425  -19.251055970 -59.440110 20.93799807  -80.71491 42.21280
## 2005.1452  -12.401040702 -52.590095 27.78801354  -73.86490 49.06281
## 2005.1479  -19.955345295 -60.144400 20.23370915  -81.41920 41.50851
## 2005.1507  -41.661244932 -81.850300 -1.47219028 -103.12510 19.80261
## 2005.1534   12.771660600 -27.417394 52.96071545  -48.69219 74.23552
## 2005.1562    2.346785367 -37.842270 42.53584042  -59.11707 63.81064
## 2005.1589  -10.630875827 -50.819931 29.55817942  -72.09473 50.83298
## 2005.1616   20.761873796 -19.427182 60.95092925  -40.70198 82.22573
## 2005.1644   12.693208407 -27.495847 52.88226406  -48.77065 74.15707
## 2005.1671  -39.783954451 -79.973010  0.40510140 -101.24781 21.67990
## 2005.1699    0.125219691 -40.063836 40.31427575  -61.33864 61.58908
## 2005.1726   19.483471594 -20.705585 59.67252785  -41.98039 80.94733
## 2005.1753    4.036089848 -36.152967 44.22514631  -57.42777 65.49995
## 2005.1781  -30.292465859 -70.481523  9.89659080  -91.75632 31.17139
## 2005.1808    2.078989234 -38.110068 42.26804610  -59.38487 63.54285
## 2005.1836   34.672236691  -5.516820 74.86129375  -26.79162 96.13610
## 2005.1863   28.730869540 -11.458188 68.91992680  -32.73299 90.19473
## 2005.1890    1.164042646 -39.025015 41.35310011  -60.29982 62.62790
## 2005.1918  -26.004463469 -66.193521 14.18459420  -87.46832 35.45940
## 2005.1945   -7.378041996 -47.567100 32.81101587  -68.84190 54.08582
## 2005.1973   -2.900796230 -43.089854 37.28826184  -64.36466 58.56306
## 2005.2000   -2.858526386 -43.047585 37.33053189  -64.32239 58.60533
## 2005.2027  -32.512216400 -72.701275  7.67684207  -93.97608 28.95164
## 2005.2055  -45.220340630 -85.409399 -5.03128196 -106.68420 16.24352
## 2005.2082    7.382217533 -32.806841 47.57127641  -54.08164 68.84608
## 2005.2110   26.843974367 -13.345085 67.03303344  -34.61989 88.30784
## 2005.2137   18.662595375 -21.526464 58.85165465  -42.80127 80.12646
## 2005.2164  -17.725392548 -57.914452 22.46366693  -79.18926 43.73847
## 2005.2192    1.646948641 -38.542111 41.83600832  -59.81691 63.11081
## 2005.2219   37.725771774  -2.463288 77.91483166  -23.73809 99.18963
## 2005.2247   21.701605209 -18.487455 61.89066529  -39.76226 83.16547
## 2005.2274   -0.046163600 -40.235224 40.14289668  -61.51003 61.41770
## 2005.2301  -15.712653958 -55.901714 24.47640653  -77.17652 45.75121
## 2005.2329  -11.008156055 -51.197217 29.18090463  -72.47202 50.45571
## 2005.2356   13.548278587 -26.640782 53.73733947  -47.91559 75.01214
## 2005.2384    4.745708874 -35.443352 44.93476996  -56.71816 66.20957
## 2005.2411  -19.765347525 -59.954409 20.42371377  -81.22921 41.69852
## 2005.2438    4.037778863 -36.151283 44.22684035  -57.42609 65.50164
## 2005.2466    5.622008468 -34.567053 45.81107016  -55.84186 67.08587
## 2005.2493   -2.019117282 -42.208179 38.16994461  -63.48298 59.44475
## 2005.2521  -30.234103589 -70.423166  9.95495851  -91.69797 31.22976
## 2005.2548  -11.838404799 -52.027467 28.35065750  -73.30227 49.62546
## 2005.2575   20.566094177 -19.622968 60.75515668  -40.89777 82.02996
## 2005.2603   29.835355832 -10.353707 70.02441853  -31.62851 91.29922
## 2005.2630   -2.148655290 -42.337718 38.04040761  -63.61252 59.31521
## 2005.2658  -11.185791576 -51.374855 29.00327153  -72.64966 50.27808
## 2005.2685  -19.836195809 -60.025259 20.35286749  -81.30006 41.62767
## 2005.2712    0.418781954 -39.770282 40.60784546  -61.04509 61.88265
## 2005.2740   11.666211928 -28.522852 51.85527563  -49.79766 73.13008
## 2005.2767   -0.073430186 -40.262494 40.11563372  -61.53730 61.39044
## 2005.2795   36.644505799  -3.544558 76.83356991  -24.81936 98.10838
## 2005.2822   -3.517731394 -43.706796 36.67133292  -64.98160 57.94614
## 2005.2849  -14.133310024 -54.322375 26.05575449  -75.59718 47.33056
## 2005.2877   -3.573322897 -43.762388 36.61574182  -65.03719 57.89055
## 2005.2904   -0.515860914 -40.704926 39.67320400  -61.97973 60.94801
## 2005.2932    5.329225908 -34.859839 45.51829102  -56.13465 66.79310
## 2005.2959  -10.149432995 -50.338498 30.03963232  -71.61330 51.31444
## 2005.2986  -13.823662748 -54.012728 26.36540277  -75.28753 47.64021
## 2005.3014    8.446314707 -31.742751 48.63538043  -53.01756 69.91019
## 2005.3041   13.844330236 -26.344736 54.03339616  -47.61954 75.30820
## 2005.3068    3.791095595 -36.397971 43.98016172  -57.67278 65.25497
## 2005.3096  -12.459446645 -52.648513 27.72961968  -73.92332 49.00443
## 2005.3123    5.684866444 -34.504200 45.87393297  -55.77901 67.14874
## 2005.3151    1.027257069 -39.161810 41.21632379  -60.43662 62.49113
## 2005.3178  -10.470796068 -50.659863 29.71827086  -71.93467 50.99308
## 2005.3205    2.538472225 -37.650595 42.72753935  -58.92540 64.00235
## 2005.3233    1.904192248 -38.284875 42.09325958  -59.55968 63.36807
## 2005.3260   -5.038257764 -45.227325 35.15080977  -66.50213 56.42562
## 2005.3288    8.294144969 -31.894923 48.48321270  -53.16973 69.75802
## 2005.3315   23.526178635 -16.662889 63.71524657  -37.93770 84.99005
## 2005.3342    2.893860553 -37.295208 43.08292869  -58.57002 64.35774
## 2005.3370  -14.869379033 -55.058447 25.31968930  -76.33326 46.59450
## 2005.3397  -21.294848694 -61.483917 18.89421984  -82.75873 40.16903
## 2005.3425   -2.922297359 -43.111366 37.26677138  -64.38617 58.54158
## 2005.3452   19.434507460 -20.754561 59.62357640  -42.02937 80.89838
## 2005.3479  -10.286921224 -50.475990 29.90214792  -71.75080 51.17696
## 2005.3507  -17.516077712 -57.705147 22.67299163  -78.97996 43.94780
## 2005.3534    4.590598236 -35.598471 44.77966778  -56.87328 66.05448
## 2005.3562    6.062425054 -34.126645 46.25149480  -55.40145 67.52630
## 2005.3589  -11.299568257 -51.488638 28.88950169  -72.76345 50.16431
## 2005.3616   -4.931292892 -45.120363 35.25777726  -66.39517 56.53259
## 2005.3644   10.999965659 -29.189105 51.18903601  -50.46391 72.46384
## 2005.3671    8.745706509 -31.443364 48.93477706  -52.71817 70.20959
## 2005.3699    4.763507401 -35.425563 44.95257815  -56.70037 66.22739
## 2005.3726   -4.202190244 -44.391261 35.98688071  -65.66607 57.26169
## 2005.3753   -0.649345017 -40.838416 39.53972614  -62.11323 60.81454
## 2005.3781    3.370809294 -36.818262 43.55988065  -58.09307 64.83469
## 2005.3808   11.633735944 -28.555336 51.82280750  -49.83014 73.09762
## 2005.3836   -7.199646116 -47.388718 32.98942564  -68.66353 54.26424
## 2005.3863   -6.187807773 -46.376880 34.00126419  -67.65169 55.27607
## 2005.3890   -6.551566742 -46.740639 33.63750542  -68.01545 54.91232
## 2005.3918    8.686917821 -31.502155 48.87599018  -52.77696 70.15080
## 2005.3945    3.947583503 -36.241489 44.13665607  -57.51630 65.41147
## 2005.3973    1.214615833 -38.974457 41.40368860  -60.24927 62.67850
## 2005.4000   -1.062335916 -41.251409 39.12673705  -62.52622 60.40155
## 2005.4027   -1.870528889 -42.059602 38.31854428  -63.33441 59.59335
## 2005.4055   -5.313753388 -45.502827 34.87531998  -66.77764 56.15013
## 2005.4082    1.686078435 -38.502995 41.87515200  -59.77781 63.14996
## 2005.4110    6.732946270 -33.456128 46.92202004  -54.73094 68.19683
## 2005.4137   -4.712038561 -44.901113 35.47703541  -66.17592 56.75185
## 2005.4164   -3.321199016 -43.510273 36.86787516  -64.78508 58.14269
## 2005.4192   11.739433482 -28.449641 51.92850786  -49.72445 73.20332
## 2005.4219    2.369557402 -37.819517 42.55863198  -59.09433 63.83344
## 2005.4247   -2.344310137 -42.533385 37.84476464  -63.80820 59.11958
## 2005.4274   -4.828056199 -45.017131 35.36101878  -66.29194 56.63583
## 2005.4301    3.804976526 -36.384099 43.99405171  -57.65891 65.26886
## 2005.4329   15.258119311 -24.930956 55.44719469  -46.20577 76.72201
## 2005.4356    4.041047614 -36.148028 44.23012320  -57.42284 65.50493
## 2005.4384  -20.267566373 -60.456642 19.92150941  -81.73145 41.19632
## 2005.4411   -7.155687836 -47.344764 33.03338815  -68.61958 54.30820
## 2005.4438    1.895808570 -38.293268 42.08488476  -59.56808 63.35970
## 2005.4466   12.439120205 -27.749956 52.62819659  -49.02477 73.90301
## 2005.4493    4.917837084 -35.271240 45.10691367  -56.54605 66.38173
## 2005.4521    4.752437341 -35.436639 44.94151413  -56.71145 66.21633
## 2005.4548   -0.918585749 -41.107663 39.27049124  -62.38247 60.54530
## 2005.4575    0.890125652 -39.298952 41.07920284  -60.57376 62.35402
## 2005.4603    0.788916716 -39.400161 40.97799411  -60.67497 62.25281
## 2005.4630   -2.224609855 -42.413687 37.96446774  -63.68850 59.23928
## 2005.4658    0.195491553 -39.993586 40.38456935  -61.26840 61.65938
## 2005.4685   10.283187554 -29.905890 50.47226555  -51.18070 71.74708
## 2005.4712   -0.590986679 -40.780065 39.59809152  -62.05488 60.87290
## 2005.4740   -4.062571494 -44.251650 36.12650691  -65.52646 57.40132
## 2005.4767   -0.740515476 -40.929594 39.44856313  -62.20441 60.72338
## 2005.4795   10.785741737 -29.403337 50.97482054  -50.67815 72.24963
## 2005.4822    6.895598254 -33.293481 47.08467726  -54.56829 68.35949
## 2005.4849   -7.876892225 -48.065971 32.31218698  -69.34078 53.58700
## 2005.4877   -8.378733784 -48.567813 31.81034562  -69.84263 53.08516
## 2005.4904   -6.135640188 -46.324720 34.05343942  -67.59953 55.32825
## 2005.4932   -3.140599854 -43.329680 37.04847996  -64.60449 58.32329
## 2005.4959    4.872664912 -35.316415 45.06174492  -56.59123 66.33656
## 2005.4986    0.471194505 -39.717886 40.66027472  -60.99270 61.93509
## 2005.5014    1.819876977 -38.369203 42.00895739  -59.64402 63.28377
## 2005.5041    0.260866590 -39.928214 40.44994721  -61.20303 61.72476
## 2005.5068    0.652872013 -39.536209 40.84195283  -60.81102 62.11677
## 2005.5096    3.158157520 -37.030923 43.34723854  -58.30574 64.62205
## 2005.5123   -4.899607996 -45.088689 35.28947322  -66.36350 56.56429
## 2005.5151  -14.255848780 -54.444930 25.93323264  -75.71974 47.20805
## 2005.5178   -5.346161444 -45.535243 34.84292018  -66.81006 56.11773
## 2005.5205   -0.130601655 -40.319683 40.05848017  -61.59450 61.33329
## 2005.5233   -1.429946630 -41.619029 38.75913539  -62.89384 60.03395
## 2005.5260   -3.520436698 -43.709519 36.66864553  -64.98433 57.94346
## 2005.5288   -4.423084011 -44.612166 35.76599842  -65.88698 57.04081
## 2005.5315    6.860193016 -33.328890 47.04927564  -54.60370 68.32409
## 2005.5342   12.305945063 -27.883138 52.49502789  -49.15795 73.76984
## 2005.5370    8.377967953 -31.811115 48.56705098  -53.08593 69.84187
## 2005.5397   -9.578199522 -49.767283 30.61088371  -71.04210 51.88570
## 2005.5425  -11.869938832 -52.059022 28.31914460  -73.33384 49.59396
## 2005.5452   -6.124843841 -46.313927 34.06423979  -67.58874 55.33906
## 2005.5479    5.153006412 -35.036077 45.34209025  -56.31089 66.61691
## 2005.5507    5.680054223 -34.509030 45.86913826  -55.78385 67.14395
## 2005.5534   -8.809371753 -48.998456 31.37971248  -70.27327 52.65453
## 2005.5562   -8.043412560 -48.232497 32.14567188  -69.50731 53.42049
## 2005.5589    6.938560688 -33.250524 47.12764533  -54.52534 68.40246
## 2005.5616    0.317952259 -39.871133 40.50703710  -61.14595 61.78185
## 2005.5644    0.589263475 -39.599822 40.77834852  -60.87464 62.05317
## 2005.5671    0.040700618 -40.148385 40.22978586  -61.42320 61.50460
## 2005.5699   -1.392503849 -41.581589 38.79658160  -62.85641 60.07140
## 2005.5726  -10.096538067 -50.285624 30.09254758  -71.56044 51.36736
## 2005.5753    1.172272123 -39.016814 41.36135797  -60.29163 62.63617
## 2005.5781    3.011391289 -37.177695 43.20047734  -58.45251 64.47529
## 2005.5808   -5.249462129 -45.438548 34.93962412  -66.71337 56.21444
## 2005.5836   -0.009372358 -40.198459 40.17971409  -61.47328 61.45453
## 2005.5863    9.973797669 -30.215289 50.16288432  -51.49011 71.43770
## 2005.5890   11.337201906 -28.851885 51.52628876  -50.12670 72.80111
## 2005.5918    1.363517866 -38.825569 41.55260492  -60.10039 62.82742
## 2005.5945   -6.679088233 -46.868175 33.50999902  -68.14299 54.78482
## 2005.5973  -10.078155251 -50.267243 30.11093221  -71.54206 51.38575
## 2005.6000   -3.494046882 -43.683135 36.69504078  -64.95795 57.96986
## 2005.6027    0.365457595 -39.823630 40.55454546  -61.09845 61.82936
## 2005.6055    1.321938136 -38.867150 41.51102620  -60.14197 62.78584
## 2005.6082    0.259100699 -39.929988 40.44818896  -61.20481 61.72301
## 2005.6110    5.211858549 -34.977230 45.40094701  -56.25205 66.67577
## 2005.6137    0.230382593 -39.958706 40.41947126  -61.23352 61.69429
## 2005.6164  -11.280004609 -51.469093 28.90908426  -72.74391 50.18390
## 2005.6192   -9.558929989 -49.748019 30.63015908  -71.02284 51.90498
## 2005.6219    8.960930222 -31.228159 49.15001949  -52.50298 70.42484
## 2005.6247   -0.781462540 -40.970552 39.40762693  -62.24537 60.68245
## 2005.6274  -10.276035813 -50.465125 29.91305386  -71.73994 51.18787
## 2005.6301  -14.015221099 -54.204311 26.17386877  -75.47913 47.44869
## 2005.6329    7.102783580 -33.086306 47.29187366  -54.36113 68.56669
## 2005.6356    4.806113672 -35.382977 44.99520395  -56.65780 66.27002
## 2005.6384  -16.413671966 -56.602762 23.77541851  -77.87758 45.05024
## 2005.6411   -0.714025543 -40.903116 39.47506514  -62.17794 60.74988
## 2005.6438   16.152691648 -24.036399 56.34178253  -45.31122 77.61660
## 2005.6466   10.212896722 -29.976194 50.40198780  -51.25101 71.67681
## 2005.6493   -6.692916168 -46.882007 33.49617512  -68.15683 54.77099
## 2005.6521   -3.369177506 -43.558269 36.81991398  -64.83309 58.09473
## 2005.6548    1.636830061 -38.552262 41.82592175  -59.82708 63.10074
## 2005.6575   12.876453072 -27.312639 53.06554496  -48.58746 74.34037
## 2005.6603    3.065502726 -37.123589 43.25459481  -58.39841 64.52942
## 2005.6630  -11.041156105 -51.230248 29.14793618  -72.50507 50.42276
## 2005.6658   -6.676852735 -46.865945 33.51223976  -68.14077 54.78706
## 2005.6685   15.775552886 -24.413540 55.96464558  -45.68836 77.23947
## 2005.6712   -6.437295515 -46.626388 33.75179738  -67.90121 55.02662
## 2005.6740   -4.519579817 -44.708673 35.66951328  -65.98349 56.94433
## 2005.6767   -0.550572890 -40.739666 39.63852041  -62.01449 60.91334
## 2005.6795  -13.364800765 -53.553894 26.82429273  -74.82872 48.09911
## 2005.6822  -23.695929400 -63.885023 16.49316430  -85.15984 37.76799
## 2005.6849   14.651046788 -25.538047 54.84014069  -46.81287 76.11496
## 2005.6877   25.348432274 -14.840662 65.53752638  -36.11548 86.81235
## 2005.6904   11.883488878 -28.305605 52.07258318  -49.58043 73.34740
## 2005.6932   -0.444430854 -40.633525 39.74466365  -61.90835 61.01949
## 2005.6959   -3.640901748 -43.829996 36.54819296  -65.10482 57.82301
## 2005.6986   -7.302176963 -47.491272 32.88691794  -68.76609 54.16174
## 2005.7014   -3.641106308 -43.830201 36.54798880  -65.10502 57.82281
## 2005.7041   -2.972993469 -43.162089 37.21610184  -64.43691 58.49092
## 2005.7068   -6.482059185 -46.671155 33.70703633  -67.94598 54.98186
## 2005.7096    3.197266485 -36.991829 43.38636220  -58.26665 64.66118
## 2005.7123    4.045585404 -36.143511 44.23468132  -57.41833 65.50950
## 2005.7151  -15.747323203 -55.936419 24.44177291  -77.21124 45.71660
## 2005.7178   -0.359501137 -40.548597 39.82959518  -61.82342 61.10442
## 2005.7205   14.939584253 -25.249512 55.12868077  -46.52433 76.40350
## 2005.7233   -0.420247627 -40.609344 39.76884909  -61.88417 61.04367
## 2005.7260   -4.405194451 -44.594291 35.78390247  -65.86911 57.05873
## 2005.7288   -7.162702058 -47.351799 33.02639506  -68.62662 54.30122
## 2005.7315   29.473710313 -10.715387 69.66280764  -31.99021 90.93763
## 2005.7342    7.289043929 -32.900054 47.47814145  -54.17488 68.75296
## 2005.7370   14.753126927 -25.435971 54.94222465  -46.71079 76.21705
## 2005.7397  -11.318599800 -51.507698 28.87049813  -72.78252 50.14532
## 2005.7425   -8.433247773 -48.622346 31.75585035  -69.89717 53.03067
## 2005.7452   -5.549215673 -45.738314 34.63988266  -67.01314 55.91471
## 2005.7479  -23.085363008 -63.274462 17.10373552  -84.54929 38.37856
## 2005.7507    2.687074838 -37.502024 42.87617357  -58.77685 64.15100
## 2005.7534   10.355443743 -29.833655 50.54454268  -51.10848 71.81937
## 2005.7562    6.474768021 -33.714331 46.66386715  -54.98916 67.93869
## 2005.7589    9.017072063 -31.172027 49.20617140  -52.44685 70.48100
## 2005.7616   -0.313049178 -40.502149 39.87605036  -61.77697 61.15087
## 2005.7644  -18.192051078 -58.381151 21.99704866  -79.65598 43.27187
## 2005.7671   -4.668004332 -44.857104 35.52109561  -66.13193 56.79592
## 2005.7699    4.242212012 -35.946888 44.43131215  -57.22171 65.70614
## 2005.7726   13.326952157 -26.862148 53.51605250  -48.13697 74.79088
## 2005.7753   11.868349688 -28.320751 52.05745023  -49.59558 73.33227
## 2005.7781   10.038751220 -30.150350 50.22785196  -51.42517 71.50268
## 2005.7808   -6.022435168 -46.211536 34.16666578  -67.48636 55.44149
## 2005.7836   -1.493373840 -41.682475 38.69572731  -62.95730 59.97055
## 2005.7863   -3.863657776 -44.052759 36.32544357  -65.32758 57.60027
## 2005.7890   28.491806026 -11.697296 68.68090758  -32.97212 89.95573
## 2005.7918   23.366433533 -16.822668 63.55553528  -38.09749 84.83036
## 2005.7945   -2.025033965 -42.214136 38.16406799  -63.48896 59.43889
## 2005.7973  -10.115874669 -50.304977 30.07322748  -71.57980 51.34805
## 2005.8000  -22.120572462 -62.309675 18.06852989  -83.58450 39.34336
## 2005.8027  -18.633097093 -58.822200 21.55600546  -80.09703 42.83083
## 2005.8055   11.472170479 -28.716932 51.66127324  -49.99176 72.93610
## 2005.8082   -4.452967296 -44.642070 35.73613566  -65.91690 57.01096
## 2005.8110   -2.088101534 -42.277205 38.10100163  -63.55203 59.37583
## 2005.8137  -30.804157815 -70.993261  9.38494555  -92.26809 30.65977
## 2005.8164    7.067409895 -33.121694 47.25651346  -54.39652 68.53134
## 2005.8192    3.712546384 -36.476557 43.90165015  -57.75138 65.17648
## 2005.8219    7.572529095 -32.616575 47.76163306  -53.89140 69.03646
## 2005.8247   28.150278060 -12.038826 68.33938223  -33.31365 89.61421
## 2005.8274  -15.603545637 -55.792650 24.58555873  -77.06748 45.86039
## 2005.8301  -16.207408604 -56.396513 23.98169596  -77.67134 45.25652
## 2005.8329  -10.645241814 -50.834347 29.54386296  -72.10917 50.81869
## 2005.8356   35.710018195  -4.479087 75.89912317  -25.75391 97.17395
## 2005.8384    3.689909279 -36.499196 43.87901445  -57.77402 65.15384
## 2005.8411  -35.408441562 -75.597547  4.78066381  -96.87237 26.05549
## 2005.8438   11.358040410 -28.831065 51.54714598  -50.10589 72.82197
## 2005.8466   38.429973769  -1.759132 78.61907955  -23.03396 99.89391
## 2005.8493   24.063058391 -16.126048 64.25216437  -37.40088 85.52699
## 2005.8521   -5.787126206 -45.976232 34.40197997  -67.25106 55.67681
## 2005.8548  -22.784474795 -62.973581 17.40463159  -84.24841 38.67946
## 2005.8575   -1.968019962 -42.157127 38.22108662  -63.43195 59.49591
## 2005.8603  -37.829400588 -78.018507  2.35970620  -99.29334 23.63453
## 2005.8630  -40.123969149 -80.313076  0.06513784 -101.58790 21.33997
## 2005.8658   20.501802312 -19.687305 60.69090950  -40.96213 81.96574
## 2005.8685   35.038623064  -5.150484 75.22773045  -26.42531 96.50256
## 2005.8712   13.309700314 -26.879407 53.49880790  -48.15424 74.77364
## 2005.8740   -4.899793566 -45.088901 35.28931422  -66.36373 56.56414
## 2005.8767  -10.019477766 -50.208586 30.16963022  -71.48341 51.44446
## 2005.8795   24.652504530 -15.536604 64.84161272  -36.81143 86.11644
## 2005.8822    4.821624372 -35.367484 45.01073277  -56.64231 66.28556
## 2005.8849   -4.247618128 -44.436727 35.94149047  -65.71156 57.21632
## 2005.8877  -12.995706907 -53.184816 27.19340189  -74.45964 48.46823
## 2005.8904   -2.966753841 -43.155863 37.22235516  -64.43069 58.49718
## 2005.8932   16.489091830 -23.700017 56.67820103  -44.97485 77.95303
## 2005.8959   -8.341499713 -48.530609 31.84760969  -69.80544 53.12244
## 2005.8986    6.585693797 -33.603416 46.77480340  -54.87825 68.04963
## 2005.9014  -21.514480942 -61.703591 18.67462886  -82.97842 39.94946
## 2005.9041    6.866791027 -33.322319 47.05590103  -54.59715 68.33073
## 2005.9068   -8.491147167 -48.680257 31.69796304  -69.95509 52.97279
## 2005.9096    7.366899570 -32.822211 47.55600998  -54.09704 68.83084
## 2005.9123  -14.278205782 -54.467316 25.91090483  -75.74215 47.18573
## 2005.9151  -29.059196359 -69.248307 11.12991445  -90.52314 32.40474
## 2005.9178   20.403320040 -19.785791 60.59243105  -41.06062 81.86726
## 2005.9205    6.991772363 -33.197339 47.18088357  -54.47217 68.45571
## 2005.9233   18.493458337 -21.695653 58.68256975  -42.97048 79.95740
## 2005.9260   18.176142134 -22.012969 58.36525375  -43.28780 79.64008
## 2005.9288   12.451507815 -27.737604 52.64061963  -49.01243 73.91545
## 2005.9315    1.251708469 -38.937404 41.44082049  -60.21223 62.71565
## 2005.9342   33.475024666  -6.714088 73.66413688  -27.98892 94.93897
## 2005.9370  -18.313315940 -58.502428 21.87579648  -79.77726 43.15063
## 2005.9397   -4.757711441 -44.946824 35.43140118  -66.22166 56.70623
## 2005.9425   -2.387925109 -42.577038 37.80118771  -63.85187 59.07602
## 2005.9452   15.986501530 -24.202611 56.17561455  -45.47744 77.45045
## 2005.9479    1.441643683 -38.747470 41.63075691  -60.02230 62.90559
## 2005.9507   -4.207474292 -44.396588 35.98163913  -65.67142 57.25647
## 2005.9534    5.818011203 -34.371102 46.00712483  -55.64593 67.28196
## 2005.9562   -5.612617543 -45.801731 34.57649629  -67.07656 55.85133
## 2005.9589    7.995900077 -32.193214 48.18501411  -53.46805 69.45985
## 2005.9616    0.870249733 -39.318864 41.05936396  -60.59370 62.33420
## 2005.9644   -1.485733515 -41.674848 38.70338092  -62.94968 59.97821
## 2005.9671    4.522164280 -35.666950 44.71127891  -56.94178 65.98611
## 2005.9699   20.927021402 -19.262093 61.11613624  -40.53693 82.39097
## 2005.9726   10.719934566 -29.469180 50.90904960  -50.74401 72.18388
## 2005.9753  -17.023367657 -57.212483 23.16574758  -78.48732 44.44058
## 2005.9781  -28.848344045 -69.037459 11.34077139  -90.31229 32.61560
## 2005.9808  -32.141652930 -72.330769  8.04746271  -93.60560 29.32230
## 2005.9836  -15.207289173 -55.396405 24.98182667  -76.67124 46.25666
## 2005.9863    2.797376773 -37.391739 42.98649282  -58.66657 64.26133
## 2005.9890    4.351510375 -35.837606 44.54062662  -57.11244 65.81546
## 2005.9918   14.270432014 -25.918684 54.45954846  -47.19352 75.73438
## 2005.9945   -2.076693282 -42.265810 38.11242336  -63.54064 59.38726
## 2005.9973   12.836645857 -27.352471 53.02576270  -48.62730 74.30060
## 2006.0000  -16.715050963 -56.904168 23.47406609  -78.17900 44.74890
## 2006.0027    3.020884290 -37.168233 43.21000154  -58.44307 64.48484
## 2006.0055    7.999234299 -32.189883 48.18835175  -53.46472 69.46319

##NOTE: AIC = 31986.86 , BIC = 32003.94, RMSE = 31.34532

Also, the headline of the plot below contains ETS(A,N,N) showing that the automated model characterized error of the “tsWSR_PK” SERIES as Additative (A), trend as None (A) and seasonality as None (N) which is equivalent to simple exponential model(ses) with only alpha parameter i.e level.

#Model 2 : ARIMA Model(AutoRegressive Moving Average) It is used for forecasting for both seasonal and non- seasonal level.

#2.1 Auto Arima model without sesonality:

set.seed(301)
model_Var11 <- auto.arima(train10, seasonal = FALSE)
summary(model_Var11)
## Series: train10 
## ARIMA(2,0,2) with zero mean 
## 
## Coefficients:
##          ar1      ar2      ma1      ma2
##       0.6267  -0.1022  -0.4235  -0.3696
## s.e.  0.0495   0.0509   0.0468   0.0491
## 
## sigma^2 estimated as 1016:  log likelihood=-10702.27
## AIC=21414.55   AICc=21414.57   BIC=21443.01
## 
## Training set error measures:
##                      ME     RMSE      MAE MPE MAPE      MASE         ACF1
## Training set -0.4976616 31.85287 22.96835 NaN  Inf 0.6275488 0.0004603258

##NOTE: AIC = 21414.55, BIC = 21443.01, RMSE = 31.85287

tsdisplay(residuals(model_Var11), lag.max = 45, main = '(1,1,1) Model Residuals')

#2.2 Auto Arima model with sesonality:

set.seed(301)
model2_Var11 <- auto.arima(train10, seasonal= TRUE)
summary(model2_Var11)
## Series: train10 
## ARIMA(4,0,2) with zero mean 
## 
## Coefficients:
##          ar1     ar2      ar3     ar4      ma1      ma2
##       0.3504  0.2212  -0.1638  0.0722  -0.1467  -0.6358
## s.e.  0.1240  0.1421   0.0764  0.0433   0.1213   0.1155
## 
## sigma^2 estimated as 1016:  log likelihood=-10701.17
## AIC=21416.34   AICc=21416.39   BIC=21456.19
## 
## Training set error measures:
##                      ME     RMSE      MAE MPE MAPE      MASE         ACF1
## Training set -0.5104428 31.83673 22.95572 NaN  Inf 0.6272036 0.0003918353

##NOTE: AIC = 21416.34, BIC = 21456.19, RMSE = 31.83673

tsdisplay(residuals(model2_Var11), lag.max = 20, main = 'Seasonal Model Residuals')

#Plotting forecast for each model

par(mfrow = c(2,2))
fit_auto_train_Var11 %>% forecast(h=341) %>% autoplot() 

model2_Var11 %>% forecast(h=341) %>% autoplot() 

model_Var11 %>% forecast(h=341) %>% autoplot() 

COMMENT:
1. The plots includes 80% to 95% confidence interval.
2. For ARIMA models it plots the forecast but because of the nature of autoregressive process of 1 lag i.e. AR1, these forecast stabilizes quite smoothly.

Plot and comment on the residuals of the fitted data for both models in the same plot.

Residual Plot - to confirm no problem with this model

par(mfrow= c(3,2))
hist(fit_auto_train_Var11$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(fit_auto_train_Var11$residuals))

hist(model_Var11$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model_Var11$residuals))

hist(model2_Var11$residuals,col = 'red',xlab = 'Error',main = 'Histogram Of Residuals',freq= FALSE)
lines(density(model2_Var11$residuals))

OBSERVATION: The residuals for each model are normally distributed, which shows the good fit of the models. It is normally distrbuted

Testing the autocorrelation and partial autocorrelation of the residuals (using the plots and Ljung-Box Q statistic) up to an appropriate lag *

#ACF & PACF PLOTS

par(mfrow = c(3,2))
acf(fit_auto_train_Var11$residuals, main = 'Correlogram')
pacf(fit_auto_train_Var11$residuals, main = 'Partial Corelogram')

acf(model_Var11$residuals, main = 'Correlogram')
pacf(model_Var11$residuals, main = 'Partial Corelogram')

acf(model2_Var11$residuals, main = 'Correlogram')
pacf(model2_Var11$residuals, main = 'Partial Corelogram')

NOTE:
1. Residuals are the difference between actaual and fitted values
2. These blue lines are signficance bounds
3. This graph shows us the autocorelations for insample forecast errors
4. Do not exceed these significance bounds for lags 1 to end

  1. Ljunx-Box Test to check the autocorelations of the residuals

This test used to test the lack of fit of a time series model.
This test is applied to the residuals of the time series model,

HO: No serial correlation upto 20 lags (does not exhibit lack of fit of the model)
Ha: Serial corelation is present (the model exhibits lack of fit)

We reject the null hypothesis, if p value is less than 0.05. This test is generally done for ARIMA models

Box.test(model_Var11$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model_Var11$residuals
## X-squared = 16.732, df = 20, p-value = 0.6703
Box.test(model2_Var11$residuals, lag =20, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  model2_Var11$residuals
## X-squared = 14.525, df = 20, p-value = 0.8029

OBSERVATIONS:
For above performed Ljunx-Box Test on ARIMA models output, we can say that p-value is more than significance level 0.05, then there is not a statistical significance.
We conclude that there is a little evidence of non-zero auto-corelations in the insample forecast errors at lags 1 to 20.
So we fail to rejects the null hypothesis. Hence, we conclude that there is no serial auto corelation present between lags. Hence, the model seems to be good fit.

Test and comment on the normality of the residuals.

Here, further testing the normality by plotting Normal Q-Q plots. From below plots we observed that most of the datapoints lies on the line. Hence, we could conclude that our model produces residuals that are normal

par(mfrow = c(2,2))
qqnorm(fit_auto_train_Var11$residuals)
qqnorm(model_Var11$residuals)
qqnorm(model2_Var11$residuals)

Validating The Model by testing the model performance with Holdout set

Here, choosing RMSE as evaluation parameter. The lower RMSE indicates a more accurate forecast.

Below, is the comparison of model. We are fitting the model to test set which is the holdout set. And checking the accuracy using RMSE(Root mean square error) as an evaluation parameter.

accuracy(forecast(fit_auto_train_Var11), test10) ["Test set", "RMSE"]
## [1] 39.2348
#o/p - 39.2348
accuracy(forecast(model_Var11), test10) ["Test set", "RMSE"]
## [1] 34.85059
#O/P - 34.85059
accuracy(forecast(model2_Var11), test10) ["Test set", "RMSE"]
## [1] 34.81429
#o/p - 34.81429
accuracy(forecast(model_Var11), test10) ["Test set", "RMSE"]
## [1] 34.85059
#O/P - 34.81429

NOTE:- Here, the RMSE value is less for ARIMA models compared to exponential model.
The minimum RMSE is of “model2_Var11”

COMMENT ON BEST MODEL AND ITS PARAMETERS

We observed the best model to be ARIMA model i.e.“model2_Var11”.

summary(model2_Var11)
## Series: train10 
## ARIMA(4,0,2) with zero mean 
## 
## Coefficients:
##          ar1     ar2      ar3     ar4      ma1      ma2
##       0.3504  0.2212  -0.1638  0.0722  -0.1467  -0.6358
## s.e.  0.1240  0.1421   0.0764  0.0433   0.1213   0.1155
## 
## sigma^2 estimated as 1016:  log likelihood=-10701.17
## AIC=21416.34   AICc=21416.39   BIC=21456.19
## 
## Training set error measures:
##                      ME     RMSE      MAE MPE MAPE      MASE         ACF1
## Training set -0.5104428 31.83673 22.95572 NaN  Inf 0.6272036 0.0003918353

For the best model ARIMA(4,0,2) ,which means

p = 4
d = 0
q = 2

For the best model, 4 members of lag are used as a predictor. As the series is stationary so members of differences needed for stationarity is 0.
2 are the number of lagged forecasts errors used in the prediction equation.
AIC = 21416.34, RMSE = 31.83673 which is minimum as compared to other models. AICc(corrected AIC) = 21416.39

REFERENCES

[1] https://cran.r-project.org/web/packages/imputeTS/vignettes/imputeTS-Time-Series-Missing-Value-Imputation-in-R.pdf
[2] https://www.kaggle.com/juejuewang/handle-missing-values-in-time-series-for-beginners
[3] https://towardsdatascience.com/how-to-handle-missing-data-8646b18db0d4
[4] https://otexts.com/fpp2/missing-outliers.html
[5] https://www.analyticsvidhya.com/blog/2018/09/non-stationary-time-series-python/
[6] https://johannesmehlem.com/blog/exponential-smoothing-time-series-forecasting-r/
[7] Lecture Notes